Local solution to an energy critical 2-D stochastic wave equation with exponential nonlinearity in a bounded domain

TL;DR

This paper establishes the local existence and uniqueness of solutions for a 2-D stochastic wave equation with exponential nonlinearity, using Strichartz inequalities and fixed point methods, and discusses potential blow-up scenarios.

Contribution

It provides the first local well-posedness result for an energy-critical stochastic wave equation with exponential nonlinearity in two dimensions.

Findings

Proved local existence and uniqueness of solutions.

Derived Strichartz inequalities for stochastic wave equations.

Identified conditions leading to solution blow-up.

Abstract

We prove the existence and the uniqueness of a local maximal solution to an $H^1$-critical stochastic wave equation with multiplicative noise on a smooth bounded domain $\mathcal{D} \subset \mathbb{R}^2$ with exponential nonlinearity. First, we derive the appropriate deterministic and stochastic Strichartz inequalities in suitable spaces and, then use them in arguments based on fixed point method to show the local well-posedness result. We also present an explosion result for the constructed unique local maximal solution.

Equations

$$u_{tt}-\Delta u=-u|u{|}^{p-1}+|u{|}^q\dot{W},\text{ s.t. }u(0)=u_0,{\partial }_{t}u(0)=u_1,$$

$$\left\{\begin{array}[]{ll}u_{tt}+Au+F(u)=G(u)\dot{W}\quad\textrm{ in }[0,\infty)\times{\mathscr{O}}\\ u(0)=u_{0},~{}~{}u_{t}(0)=u_{1}\quad\textrm{ on }{\mathscr{O}},\end{array}\right.$$

$$\Vert u{\Vert }_{L^q(\mathcal{O})}:={\left({\int }_{\mathcal{O}}|u(x){|}^{q}dx\right)}^{\frac{1}{q}},u\in L^q(\mathcal{O}).$$

$$\Vert u{\Vert }_{L^{\infty }(\mathcal{O})}:=\mathrm{esssup}\{|u(x)|:x\in \mathcal{O}\},u\in L^{\infty }(\mathcal{O}).$$

$$\|u\|_{\mathrm{C}([0,T];H)}:=\sup_{t\in[0,T]}\|u(t)\|_{H},\quad u\in\mathrm{C}\bigl{(}[0,T];H\bigr{)}.$$

$$\Vert u{\Vert }_{L^p(0,T;E)}:={\left({\int }_0^T\Vert u(t){\Vert }_E^{p}dt\right)}^{\frac{1}{p}},u\in L^p(0,T;E),$$

$$\Vert u{\Vert }_{{\dot{\mathrm{C}}}^{\alpha }(\mathcal{O})}:=\underset{x\ne y}{\sup }\frac{|u(x)-u(y)|}{|x-y{|}^{\alpha }},$$

$$\Vert f{\Vert }_{H^{s,q}(\mathcal{O})}:=\underset{\begin{array}{c}g\upharpoonright \mathcal{O}=f\\ g\in H^{s,q}({\mathbb{R}}^2)\end{array}}{\inf }\Vert g{\Vert }_{H^{s,q}({\mathbb{R}}^2)},f\in H^{s,q}(\mathcal{O}).$$

$$\mathcal{D}(-{\Delta }_D)=H^{2,2}(\mathcal{O})\cap H_0^{1,2}(\mathcal{O}),$$

$$\mathcal{D}(-{\Delta }_N)=\left\{f\in H^{2,2}(\mathcal{O}):{\partial }_{\nu }f\upharpoonright \partial \mathcal{O}=0\right\}.$$

$$-{\Delta }_D{e}_j={\lambda }_j^2{e}_j;e_j\in \mathcal{D}(-{\Delta }_D),\forall j\ge 1\text{; }0<{\lambda }_1^2\le {\lambda }_2^2\le \cdots \text{ and }{\lambda }_n^2\underset{n\to \infty }{\to }\infty .$$

$$-{\Delta }_N{e}_j$$

$$\text{ and }0={\lambda }_1^2={\lambda }_2^2=\cdots ={\lambda }_{m_0}^2<{\lambda }_{m_0+1}^2\le {\lambda }_{m_0+2}^2\le \cdots ,$$

$$\Vert u{\Vert }_{\mathcal{D}(A^{s/2})}$$

$$\mathcal{D}(A^{s/2})={\left[L^2(\mathcal{O}),\mathcal{D}(A)\right]}_{s/2}.$$

$$\mathcal{D}(A_{D,q})=H^{2,q}(\mathcal{O})\cap H_0^{1,q}(\mathcal{O}),$$

$$\mathcal{D}(A_{N,q})=\left\{f\in H^{2,q}(\mathcal{O}):{\partial }_{\nu }f\upharpoonright \partial \mathcal{O}=0\right\}.$$

$$B_{j}f(x)=\sum _{|\alpha |\le m_j}{b}_{j,\alpha }(x)D^{\alpha }f(x),b_{j,\alpha }\in {\mathrm{C}}^{\infty }(\partial \mathcal{O}).$$

$$0\le m_1<m_2<\cdots <m_k,$$

$$\sum _{|\alpha |=m_j}{b}_{j,\alpha }(x){\nu }_x^{\alpha }\ne 0,j=1,\dots ,k,$$

$$H_{\{B_j\}}^{s,q}(\mathcal{O}):=\left\{f\in H^{s,q}(\mathcal{O}):B_{j}f\upharpoonright \partial \mathcal{O}=0\text{ whenever }{m}_j<s-\frac{1}{q}\right\}.$$

$$H_D^{s,q}(\mathcal{O}):=\left\{f\in H^{s,q}(\mathcal{O}):f\upharpoonright \partial \mathcal{O}=0\text{ if }s>\frac{1}{q}\right\},$$

$$H_N^{s,q}(\mathcal{O}):=\left\{f\in H^{s,q}(\mathcal{O}):{\nu }_x\cdot \nabla f\upharpoonright \partial \mathcal{O}=0\text{ if }s>1+\frac{1}{q}\right\}.$$

$$H_N^{s,q}(\mathcal{O})=\mathcal{D}(A_{N,q}^{s/2}).$$

$$H_D^{s,q}(\mathcal{O})=\mathcal{D}(A_{D,q}^{s/2}).$$

$$\mathcal{D}(\sqrt{-{\Delta }_D})=H_0^{1,2}(\mathcal{O})\text{ and }\mathcal{D}(\sqrt{-{\Delta }_N})=H^{1,2}(\mathcal{O}).$$

$$\|X\|_{L^{p}(\Omega)}=\bigl{(}\mathbb{E}\left[\|X\|_{E}^{p}\right]\bigr{)}^{\frac{1}{p}},\quad X\in L^{p}(\Omega,\mathscr{F},\mathbb{P};E),$$

$$\|\Lambda\|_{\gamma(K,E)}:=\left(\mathbb{E}\bigg{\|}\sum_{j\in\mathbb{N}}\beta_{j}\Lambda(f_{j})\bigg{\|}_{E}^{2}\right)^{\frac{1}{2}}.$$

$$\mathrm{strong}\mathrm{top}(\gamma(K,E)):=\bigl{\{}\{L\in\gamma(K,E):Lk\in A\},\;\;k\in K,\;A\in\mathrm{top}(E)\bigr{\}},$$

$$\mathcal{B}(\gamma (K,E))=\mathcal{S}(\gamma (K,E)).$$

$${\int }_Z|f(z){|}_X^p\mu (dz)<\infty .$$

$${\int }_Z|f(z){|}_{\gamma (K,E)}^p\mu (dz)<\infty .$$

Full text

1. 1 Introduction

2. 2 Notation and conventions

3. 2.1 Function spaces and interpolation theory

4. 2.2 Stochastic analysis

5. 2.3 Various types of measurability

6. 2.4 Stopping times

7. 3 Inhomogeneous Strichartz estimates

8. 4 Stochastic Strichartz estimates

9. 4.1 Main assumptions

10. 4.2 Martingales

11. 4.3 Burkholder inequality

12. 5 Local well-posedness - preliminary results

13. 5.1 SNLWE and assumptions

14. 5.2 Definition of a local mild solution

15. 6 The main results

16. 6.1 Statements of the results on the existence and the uniqueness of solutions

17. 6.2 A proof of the existence of a local solution

18. 6.3 A proof of the uniqueness of a local solution

19. 6.4 Proof of the existence of local maximal solution

20. 6.5 Explosion for a local maximal solution

21. A Stopped processes

22. B On pointwise evaluation

23. C About the definition of a solution

24. D Stochastic Gronwall lemma

25. E Extensions of functions

1. Introduction

In this paper we consider a nonlinear wave equation subject to random forcing, called the stochastic nonlinear wave equation (SNLWE). Due to its numerous applications to physics, relativistic quantum mechanics and oceanography, SNLWEs have been thoroughly studied under various sets of assumptions, see for example [11]-[12], [19]-[28], [42]-[52], [54]-[57], [61]-[62] and references therein. The case that has so far attracted the most attention seems to be of the stochastic wave equation with initial data belonging to the energy space $H^{1}(\mathbb{R}^{d})\times L^{2}(\mathbb{R}^{d})$. For such equations, the nonlinearities can be of polynomial type, for instance the following SNLWE

[TABLE]

with the suitable exponents $p,q\in(0,\infty)$; see a series of papers by Ondreját [52], [54]-[57]. Another extensively studied important case is when the initial data $(u_{0},u_{1})$ is in $L^{2}(\mathbb{R}^{d})\times H^{-1}(\mathbb{R}^{d})$ (possibly with weights), see [61], [62] for more details. Similar problems on a bounded domain have been investigated in [14], [25] and [52].

In the deterministic case, see for instance [67], the question of solvability of (1.1) without noise, when the initial data belongs to $H^{1}(\mathbb{R}^{d})\times L^{2}(\mathbb{R}^{d})$, has been investigated in the following three cases: (i) subcritical, i.e. $p<p_{c}$; (ii) critical, i.e. $p=p_{c}$; and (iii) supercritical, i.e. $p>p_{c}$, where $p_{c}=\frac{d+2}{d-2}$. In particular, for $d=2$, any polynomial nonlinearity is subcritical. Therefore, an exponential nonlinearity is a legitimate choice for a critical one. Nonlinearities of exponential type have been studied in many physical models, e.g. a model of self-trapped beams in plasma, see [44], and mathematically in [3], [23], [40]-[41] and [51]. With the help of suitable Strichartz estimates, the existence of global solutions have been proved, in [40]-[41], in the cases when the initial energy is strictly below or at the threshold given by the sharp Moser-Trudinger inequality. Moreover, an instability result has been shown when the initial energy is strictly above the threshold.

Our aim here is to extend the existing studies to the wave equation with exponential nonlinearity subject to randomness. In this way, we generalise the above mentioned results of Ondreját for two dimensional domains, by allowing the exponential nonlinearites, as well as the results of Ibrahim, Majdoub, and Masmoudi and others by allowing randomness. To be precise, we are interested in the following stochastic nonlinear wave equation on a smooth bounded domain ${\mathscr{O}}\subset\mathbb{R}^{2}$,

[TABLE]

where $A$ is either $-\Delta_{D}$ or $-\Delta_{N}$, i.e. $-A$ is the Laplace-Beltrami operator with the Dirichlet or the Neumann boundary conditions, respectively; $(u_{0},u_{1})\in\mathscr{D}(A^{\frac{1}{2}})\times L^{2}({\mathscr{O}})$; $W=\{W(t):t\geq 0\}$ is a $K$-cylindrical Wiener process for a real separable Hilbert space $K$; $F$ and $G$ are locally Lipschitz maps with some growth properties. In particular, the functions $F(u)$ and $G(u)$ are allowed to be of the form \pm u\bigl{(}e^{4\pi u^{2}}-1\bigr{)} and hence our results cover the recent results obtained in [41]. Detailed and precise assumptions on the model are stated in the subsequent sections. We would like to stress that, to the best of our knowledge, the present paper is the first one to study the wave equations in two dimensional domain with an exponential nonlinearity and an additive or multiplicative noise. As compared with the deterministic paper [41] we prove a counterpart of its Theorem 1.4, i.e. we prove the existence of a unique local solution with a smallness condition on the gradient of initial position. The difficulty we encounter in the present paper is that although the nonlinear function is defined on the whole space $\mathscr{D}(A^{\frac{1}{2}})$, it’s values belong to the space $L^{2}$ only for the elements of the space $\mathscr{D}(A^{\frac{1}{2}})$ whose norm is sufficiently small. We emphasize that our proof of stochastic Strichartz estimate, see Theorem 4.7, simplifies and clarifies the one from [11]. Since the proof of the existence and the uniqueness presented here is obtained by means of appropriate Strichartz estimates and these estimate are different for the full domain case, we will address the question of solvability of (1.2) on $\mathbb{R}^{2}$ in a forthcoming paper.

The organization of the present paper is as follows. In Section 2, we introduce our notation and provide the required definitions used in the paper. In Sections 3 and 4, we derive the required inhomogeneous and stochastic Strichartz estimates, respectively, by the methods introduced in [17]-[18] and [11]. Section 5 is devoted to the estimates which are sufficient to apply the Banach Fixed Point Theorem in a suitable space. The proof of the existence and uniqueness of a local maximal solution is given in Section 6. Here we also formulate the results about the explosion for the constructed unique local maximal solution. In Appendix A, we provide a rigorous justification of our definition of a local mild solution. In Appendix B, we formulate a result about pointwise evaluation of $L^{p}$-valued Bochner integrals. In Appendix C we state, without proof, an equivalence of two natural definitions of a mild solution for stochastic PDE (1.2). We conclude the paper with Appendix D and E in which we prove, respectively, a slight simplification of the stochastic Gronwall Lemma [36, Lemma 5.3] and a generalization of an existence of a Lipschitz extension result [5, Corollary 3].

2. Notation and conventions

In this section we introduce notation and some basic estimates that we use throughout the paper. We write $a\lesssim b$ if there exists a universal constant $c>0$, independent of $a,b$, such that $a\leq cb$, and we write $a\simeq b$ when $a\lesssim b$ and $b\lesssim a$. In case we want to emphasize the dependence of $c$ on some parameters $a_{1},\ldots,a_{k}$, then we write, respectively, $\lesssim_{a_{1},\ldots,a_{k}}$ and $\simeq_{a_{1},\ldots,a_{k}}$. For any two Banach spaces $X,Y$, we denote by $\mathcal{L}(X,Y)$ the space of linear bounded operators $L:X\to Y$.

To state the definitions of required spaces here, we denote by $E$ and $H$ a separable Banach and Hilbert space, respectively.

2.1. Function spaces and interpolation theory

In the next few basic definitions and remarks, which are included here for the reader’s convenience, from function spaces and interpolation theory we borrow the notation from [69].

By $L^{q}({\mathscr{O}})$, for $q\in[1,\infty)$ and a bounded smooth domain ${\mathscr{O}}$ of $\mathbb{R}^{2}$, we denote the classical real Banach space of all (equivalence classes of) $\mathbb{R}$-valued $q$-integrable Lebesgue measurable functions on ${\mathscr{O}}$. The norm in $L^{q}({\mathscr{O}})$ is given by

[TABLE]

By $L^{\infty}({\mathscr{O}})$ we denote the real Banach space of all (equivalence classes of) Lebesgue measurable essentially bounded $\mathbb{R}$-valued functions defined on ${\mathscr{O}}$ with the norm

[TABLE]

For any $T>0$, the space \mathrm{C}\bigl{(}[0,T];H\bigr{)} of all $H$-valued continuous functions $u:[0,T]\to H$ endowed with the norm

[TABLE]

is the real Banach space.

We also define, for any $p\in[1,\infty)$, L^{p}\bigl{(}0,T;E\bigr{)} as the vector space of all (equivalence classes of) $E$-valued strongly measurable functions $u:[0,T]\to E$ such that $\int_{0}^{T}\|u(t)\|_{E}^{p}\,dt<\infty$. The space L^{p}\bigl{(}0,T;E\bigr{)} endowed with the norm

[TABLE]

is a real Banach space.

For $\alpha\in(0,1)$, by $\dot{\mathrm{C}}^{\alpha}({\mathscr{O}})$, the homogeneous $\alpha$-Hölder space, we mean the set of continuous functions $u$ whose Hölder coefficient

[TABLE]

is finite. Note that the Hölder coefficient serves as a seminorm. The inhomogeneous $\alpha$-Hölder space is $\mathrm{C}^{\alpha}({\mathscr{O}})=\dot{\mathrm{C}}^{\alpha}({\mathscr{O}})\cap L^{\infty}({\mathscr{O}})$ endowed with the norm $\|u\|_{\dot{\mathrm{C}}^{\alpha}({\mathscr{O}})}+\|u\|_{L^{\infty}({\mathscr{O}})}$.

For any $s\in\mathbb{R}$ and $q\in(1,\infty)$, the Sobolev space $H^{s,q}({\mathscr{O}})$ is defined as the restriction of $H^{s,q}(\mathbb{R}^{2})$ (see e.g. [69, Definition 2.3.1/1]) to ${\mathscr{O}}$ with norm

[TABLE]

Here $g\upharpoonright{\mathscr{O}}$ is the restriction in the sense of distribution. We denote the closure of $\mathrm{C}_{0}^{\infty}({\mathscr{O}})$, the set of smooth functions defined over ${\mathscr{O}}$ with compact support, in $H^{s,q}({\mathscr{O}})$ by ${H}_{0}^{s,q}({\mathscr{O}})$.

Throughout the whole paper, we denote by $A$ the Dirichlet or the Neumann$-$Laplacian on Hilbert space $L^{2}({\mathscr{O}})$ with domain, respectively,

[TABLE]

Here $\nu$ denotes the outward normal unit vector to $\partial{\mathscr{O}}$. It is well-known, see e.g. [65], that the Dirichlet Laplacian $(-\Delta_{D},\mathscr{D}(-\Delta_{D}))$ is a positive self-adjoint operator on $L^{2}({\mathscr{O}})$ and there exists an orthonormal basis $\{e_{j}\}_{j\in\mathbb{N}}$ of $L^{2}({\mathscr{O}})$ which consists of eigenvectors of $-\Delta_{D}$. If we denote the corresponding eigenvalues by $\{\lambda_{j}^{2}\}_{j\in\mathbb{N}}$, then we have

[TABLE]

In the case of the Neumann Laplacian, $(-\Delta_{N},\mathscr{D}(-\Delta_{N}))$ is a non-negative self-adjoint operator on $L^{2}({\mathscr{O}})$ and there exists an orthonormal basis $\{e_{j}\}_{j\in\mathbb{N}}$ of $L^{2}({\mathscr{O}})$ which consists of eigenvectors of $-\Delta_{N}$. Moreover, if we denote the corresponding eigenvalues by $\{\lambda_{j}^{2}\}_{j\in\mathbb{N}}$, then we have

[TABLE]

for some $m_{0}\in\mathbb{N}$. Since we work with both the operators simultaneously, we denote the pair of operator and its domain by $(A,\mathscr{D}(A))$ and make the distinction where required.

From the functional calculus of self-adjoint operators, see for instance [71], it is known that, the power $A^{s}$ of operator $A$, for every $s\in\mathbb{R}$, is well-defined and self-adjoint. It is also known that, for any $s\in\mathbb{R}$, $\mathscr{D}(A^{s/2})$, where $A=-\Delta_{D}$ or $A=-\Delta_{N}$, with the following norm

[TABLE]

is a Hilbert space. For $s\in(0,2)$ the space $\mathscr{D}(A^{s/2})$ is equal to the following complex interpolating space, refer [69, 2.5.3/(13)],

[TABLE]

To derive the Strichartz estimate in suitable spaces, we also need to consider the Dirichlet or the Neumann$-$Laplacian on Banach space $L^{q}({\mathscr{O}})$, $q\in(1,\infty)$, denoted by $A_{D,q}$ and respectively $A_{N,q}$, with domains, respectively,

[TABLE]

Note that $A_{D,2}=-\Delta_{D}$ and $A_{N,2}=-\Delta_{N}$.

Under some reasonable assumptions on the regularity of the domain ${\mathscr{O}}$, one can show that both of these operators have very nice analytic properties. In particular both have bounded imaginary powers with exponent strictly less than $\frac{\pi}{2}$ (and thus both $-A_{D,q}$ and $-A_{N,q}$ generate analytic semigroups on the space $L^{q}({\mathscr{O}})$). As in [69], one can define the fractional powers $A_{B,q}^{r/2}$, where as below $B=D$ or $B=N$. The domains $\mathscr{D}(A_{B,q}^{r/2})$ of these operators can be identified as certain subsets of the Sobolev spaces $H^{r,q}({\mathscr{O}})$, see Lemma 2.2 below.

Next, we fix the notation for the required subspaces of $H^{s,q}({\mathscr{O}})$ which are determined by differential operators. Fix $k\in\mathbb{N}$ and let $B_{j}$, $j=1,\ldots,k$, be differential operators on $\partial{\mathscr{O}}$ defined by

[TABLE]

Then $\{B_{j}\}_{j=1}^{k}$ is said to be a normal system if and only if

[TABLE]

and for every vector $\nu_{x}$ which is normal to $\partial{\mathscr{O}}$ at $x$ the following holds

[TABLE]

where for $\alpha\in\mathbb{N}^{d}$ and $y\in\mathbb{R}^{d}$, $y^{\alpha}=\Pi_{i}y_{i}^{\alpha_{i}}$.

Definition 2.1**.**

Let $\{B_{j}\}_{j=1}^{k}$ be a normal system as defined above for some $k\in\mathbb{N}$. For $s>0,q\in(1,\infty)$, we set

[TABLE]

By taking the suitable choice of normal system $\{B_{j}\}$ in the Definition 2.1, for $s>0$ and $q\in(1,\infty)$, we define

[TABLE]

and

[TABLE]

Since the ${H}_{0}^{1,q}({\mathscr{O}})$ space can also be defined by using the $f\upharpoonright\partial{\mathscr{O}}=0$ condition appearing in (2.2) and the Neumann boundary condition appearing in (2.3) can be written as $\nu_{x}\cdot\nabla f\upharpoonright\partial{\mathscr{O}}=0$, we expect to have some relation between the spaces $H_{B}^{s,q}({\mathscr{O}})$ and $\mathscr{D}(A_{B,q}^{s/2})$ where $A=-\Delta_{B}$ with $B=D$ or $B=N$. The next stated result, which is standard in the theory of interpolation spaces, see [69, Theorem 4.3.3], provides a suitable range of $s$ for which the function spaces $H_{B}^{s,q}({\mathscr{O}})$ and $\mathscr{D}(A_{B,q}^{s/2})$ are equivalent.

Lemma 2.2**.**

With our notation from this section, we have the following

1. (1)

For $s\in(0,2)\setminus\left\{1+\frac{1}{q}\right\}$,

[TABLE] 2. (2)

For $s\in(0,2)\setminus\left\{\frac{1}{q}\right\}$,

[TABLE]

We close this subsection with the following well-known identity

[TABLE]

2.2. Stochastic analysis

Now we state a few required definitions from the theory of stochastic analysis, refer [4] and [15] for more details. Throughout the whole paper we assume that $(\Omega,\mathscr{F},\mathbb{P},\mathbb{F})$, where $\mathbb{F}:=\{\mathscr{F}_{t}:t\geq 0\}$, is a filtered probability space which satisfies the usual hypothesis, that is, the filtration $\mathbb{F}$ is right continuous and the $\sigma$-field $\mathscr{F}_{0}$ contains all $\mathbb{P}$-null sets of $\mathscr{F}$, see [47, Definition I.1.1].

As the noise, we consider a cylindrical $\mathbb{F}$-Wiener process on a real separable Hilbert space $K$, see [15, Definition 4.1]. Let us recall that $E$ is a separable Banach space. We denote by $L^{p}(\Omega,\mathscr{F},\mathbb{P};E)$, for $p\in[1,\infty)$, the Banach space of all (equivalence classes of) $E$-valued random variables equipped with the norm

[TABLE]

where $\mathbb{E}$ is the expectation operator w.r.t. $\mathbb{P}$.

Definition 2.3**.**

For any $K$, a separable Hilbert space, the set $\gamma(K,E)$ of all $\gamma$-radonifying operators consists of all bounded operators $\Lambda:K\to E$ such that the series $\sum_{j=1}^{\infty}\beta_{j}\Lambda(f_{j})$ converges in $L^{2}(\Omega,\mathscr{F},\mathbb{P};E)$ for some (or any) orthonormal basis $\{f_{j}\}_{j\in\mathbb{N}}$ of $K$ and some (or any) sequence $\{\beta_{j}\}_{j\in\mathbb{N}}$ of i.i.d. $N(0,1)$ real random variables on probability space $(\Omega,\mathscr{F},\mathbb{P})$. We set

[TABLE]

One may prove that $\|\cdot\|_{\gamma(K,E)}$ is a norm, and $(\gamma(K,E),\|\cdot\|_{\gamma(K,E)})$ is a separable Banach space, see [50, Section 4 in Chapter 1]. Note that if $K=\mathbb{R}$, then $\gamma(\mathbb{R},E)$ can be identified with $E$.

2.3. Various types of measurability

Before we continue we need to recall some basic definitions about various notions of measurability. For this purpose let us fix $(Z,\mathscr{Z})$ be a measurable space and $X$ be a Banach space. Let us denote the Borel $\sigma$-algebra of $X$ by $\mathscr{B}(X)$. The names we use are slightly untypical, since we use the letter $\mathscr{Z}$ to underline the $\sigma$-field in question.

Definition 2.4**.**

[38*, Sec 1.1., p. 2]** Suppose that $\mathscr{X}$ is a $\sigma$-field of subsets of $X$. A function $f:Z\to X$ is $\mathscr{Z}$/$\mathscr{X}$ measurable, iff the pre-image $f^{-1}(B)$ belongs to $\mathscr{Z}$ for every set $B\in\mathscr{X}$.

In particular, a function $f:Z\to X$ is $\mathscr{Z}$-Borel iff it is $\mathscr{Z}$/$\mathscr{B}(X)$ measurable.*

Definition 2.5**.**

[38, Defn 1.1.3]** A function $f:Z\to X$ is called $\mathscr{Z}$-simple iff it is of the form $f=\sum_{i=1}^{N}\mathds{1}_{A_{i}}\otimes x_{i}$ for some $N\in\mathbb{N}$, $A_{i}\in\mathscr{Z}$ and $x_{i}\in X$ for all $i=1,\ldots,N$. Here $(\mathds{1}_{A_{n}}\otimes x_{n})(s)=\mathds{1}_{A_{n}}(s)x_{n}$.

Definition 2.6**.**

[38, Defn 1.1.4]** A function $f:Z\to X$ is said to be strongly $\mathscr{Z}$-measurable iff there exists a sequence of $\mathscr{Z}$-simple functions $f_{n}:Z\to X$ such that $\lim\limits_{n\to\infty}f_{n}=f$ pointwise on $Z$.

Remark 2.7**.**

[38, Corollary 1.1.10]** If $X$ is a separable Banach space, then a function $f:Z\to X$ is strongly $\mathscr{Z}$-measurable iff it is $\mathscr{Z}$/$\mathscr{B}(X)$ measurable.

When $X$ is a space of linear operators, the situation becomes more involved. In particular, the pronoun “strongly” can be used in two different ways. Hence we have to be careful. It is well-known, see [50, Lemma 44, p. 51], that many classical $\sigma$-fields of subsets of the space $\gamma(K,E)$ coincide. In particular, the Borel $\sigma$-field $\mathscr{B}(\gamma(K,E))$, i.e. the $\sigma$-field generated by $\mathrm{top}(\gamma(K,E))$, i.e. the topology on $\gamma(K,E)$ which is induced by the norm (2.4) on $\gamma(K,E)$, is equal to the strong $\sigma$-field $\mathcal{S}(\gamma(K,E))$, i.e. the $\sigma$-field generated by the following family of strong open sets

[TABLE]

where $\mathrm{top}(E)$ is the topology on $E$ which is induced by the norm on $E$. In other words,

[TABLE]

To formulate our main result in this part of our work we need the following definitions of strong and Borel measurability of an $\gamma(K,E)$-valued function.

Definition 2.8**.**

[38, Defn 1.1.27]** A function $f:Z\to\gamma(K,E)$ is called $\mathscr{Z}$-double-strongly measurable if for all $k\in K$ the $E$-valued function $f(\cdot)k:Z\ni z\mapsto f(z)(k)\in E$ is strongly $\mathscr{Z}$-measurable according to Definition 2.6.

Definition 2.9**.**

A function $f:Z\to\gamma(K,E)$ is called strongly $\mathscr{Z}$-Borel measurable if for all $k\in K$ the $E$-valued function $f(\cdot)k:Z\ni z\mapsto f(z)(k)\in E$ is $\mathscr{Z}$-Borel according to Definition 2.4.

Proposition 2.10**.**

For a function $f:Z\to\gamma(K,E)$ the following conditions are equivalent.

• (i)

$f$* is $\mathcal{Z}$/$\mathscr{B}(\gamma(K,E))$ measurable, see Definition 2.4;*

• (ii)

$f$* is strongly $\mathscr{Z}$-measurable, see Definition 2.6;*

• (iii)

$f$* is strongly $\mathscr{Z}$-Borel measurable, see Definition 2.9;*

• (iv)

$f$* is $\mathscr{Z}$-double strongly measurable, see Definition 2.8;*

• (v)

$f$* is $\mathcal{Z}$/$\mathcal{S}(\gamma(K,E))$ measurable, see Definition 2.4.*

It is well-known, that such a result does not hold if the space $\gamma(K,E)$ is replaced by the space $\mathcal{L}(K,E)$ of all bounded and linear operators from $K$ to $E$.

Proof of Proposition 2.10.

Since $X:=\gamma(K,E)$ and $E$ are separable Banach spaces, by Remark 2.7 we infer that (i)$\Leftrightarrow$ (ii) and (iii)$\Leftrightarrow$ (iv). Moreover, (iv)$\Leftrightarrow$ (v). For this, let us observe that $k\in K$, then $f(\cdot)k=f\circ i_{k}$, where $i_{k}:\gamma(K,E)\ni L\mapsto L(k)\in E$ is the evaluation map. Thus, for every $A\in\mathrm{top}(E)$ and every $k\in K$, $f^{-1}(\{L\in\gamma(K,E):L\circ i_{k}\in A\})=(f(\cdot)k)^{-1}(A)$. Hence, the equivalence (iv)$\Leftrightarrow$ (v) follows. We conclude the proof by observing that the equivalence (i)$\Leftrightarrow$ (v) is a consequence of Neidhardt’s [50] result (2.6). This completes the proof. ∎

Let us recall that the progressive $\sigma$-field $\mathscr{B}\mathbb{F}(\mathbb{R}_{+}\times\Omega)$ consists of all sets $A\subseteq\mathbb{R}_{+}\times\Omega$ such that for any $t\in\mathbb{R}_{+}$, the set $A\cap([0,t]\times\Omega)\in\mathscr{B}([0,t])\otimes\mathscr{F}_{t}$, see [64, Definition I.4.7] and [73, 6.0.4 and 6.0.5]. Similarly we can define the $\sigma$-field $\mathscr{B}\mathbb{F}([0,T]\times\Omega)$.

Specializing Proposition 2.10 to $Z=[0,T]\times\Omega$ and $\mathcal{Z}=\mathscr{B}\mathbb{F}([0,T]\times\Omega)$, we infer that all five possible definitions of $\mathscr{B}\mathbb{F}([0,T]\times\Omega)$-measurability of processes taking values in $\gamma(K,E)$ are equivalent. In what follows we will simply use words “a progressively measurable $\gamma(K,E)$-valued process”.

Let us point out that there are even more versions of this definition. For instance, a process $\eta:[0,T]\times\Omega\to\gamma(K,E)$ is progressively Borel measurable iff $\eta^{-1}(A)\in\mathscr{B}\mathbb{F}([0,T]\times\Omega)$ for every $A\in\mathscr{B}(\gamma(K,E))$.

In a more general case, when $\gamma(K,E)$ is replaced by a separable Banach space $X$, we can only use Remark 2.7 to assert that a process $\xi:[0,T]\times\Omega\to X$ is Borel progressively measurable iff it is strongly progressively measurable.

In addition to measurable and strongly measurable functions, one can consider a notion of strongly measurable functions with respect to a measure. More precisely, we have, see [72] and [38, Definitions 1.1.13 and 1.1.14].

Definition 2.11**.**

Assume that $(Z,\mathcal{Z},\mu)$ is a measure space and $X$ is a Banach space.

• (i)

A function $f:Z\to X$ is called $\mu$-simple iff it is of the form $f=\sum_{i=1}^{N}\mathds{1}_{A_{i}}\otimes x_{i}$ for some $N\in\mathbb{N}$, $A_{i}\in\mathscr{Z}$ with $\mu(A_{i})<\infty$ and $x_{i}\in X$ for all $i=1,\ldots,N$.

• (ii)

A function $f:Z\to X$ is said to be strongly $\mu$-measurable iff there exists a sequence of $\mu$-simple functions $f_{n}:Z\to X$ such that $f_{n}\to f$, $\mu$-almost everywhere.

The next result is borrowed from [38, Proposition 1.1.16 and Remark 1.1.18].

Proposition 2.12**.**

Assume that $(Z,\mathcal{Z},\mu)$ is a $\sigma$-finite measure space and $X$ is a separable Banach space. Then the following three assertions are equivalent.

• (i)

$f$* is strongly $\mu$-measurable;*

• (ii)

there exists a strongly $\mathcal{Z}$-measurable function $\tilde{f}:Z\to X$ such that $f=\tilde{f}$, $\mu$-almost everywhere;

• (iii)

$f$* is strongly $\mathcal{Z}_{\mu}$-measurable, where $\mathcal{Z}_{\mu}$ is the completion of $\mathcal{Z}$ with respect to ${\mu}$.*

Let us also recall the following definition of Bochner spaces.

Definition 2.13**.**

Assume that $(Z,\mathcal{Z},\mu)$ is a measure space, $X$ is a Banach space and $p\in[1,\infty)$. By $\mathcal{L}^{p}(Z,\mathcal{Z};X)=\mathcal{L}^{p}(Z;X)$ we denote the vector space of all strongly $\mu$-measurable functions $f:Z\to X$ such that

[TABLE]

*By $\tilde{\mathcal{L}}^{p}(Z,\mathcal{Z};X)=\tilde{\mathcal{L}}^{p}(Z;X)$ we denote the vector space of all strongly $\mathcal{Z}$-measurable functions $f:Z\to X$ such that condition (2.7) holds.

By ${L}^{p}(Z,\mathcal{Z};X)={L}^{p}(Z;X)$ we denote the vector space of all equivalence classes of functions from $\mathcal{L}^{p}(Z;X)$. By $\tilde{{L}}^{p}(Z,\mathcal{Z};X)=\tilde{{L}}^{p}(Z;X)$ we denote the vector space of all equivalence classes of functions from $\tilde{\mathcal{L}}^{p}(Z;X)$.*

Proposition 2.12 implies that the spaces ${L}^{p}(Z,\mathcal{Z};X)$ and $\tilde{{L}}^{p}(Z,\mathcal{Z};X)$ are naturally isometrically isomorphic. In particular, for every element $g$ of ${L}^{p}(Z,\mathcal{Z};X)$, there exists a strongly $\mathcal{Z}$-measurable function $\tilde{f}:Z\to X$ such that $[\tilde{f}]=g$. Endowed with the classical norm, the Bochner space ${L}^{p}(Z;X)$ is a Banach space.

When $X=\gamma(K,E)$, where $K$ and $E$ are separable Hilbert and, respectively, Banach spaces, it follows from the above and Proposition 2.10 that the space ${L}^{p}(Z,\mathcal{Z};X)$ can be defined as the space of all equivalence classes of $\mathcal{Z}$/$\mathscr{B}(\gamma(K,E))$-measurable functions such that

[TABLE]

In fact, the $\mathcal{Z}$/$\mathscr{B}(\gamma(K,E))$-measurability can be replaced by each of the five versions of measurability listed in Proposition 2.10.

2.4. Stopping times

In this section, as throughout the whole paper, we assume that $(\Omega,\mathscr{F},\mathbb{P},\mathbb{F})$, where $\mathbb{F}:=\{\mathscr{F}_{t}:t\geq 0\}$, is a filtered probability space which satisfies the usual hypothesis, that is, the filtration $\mathbb{F}$ is right continuous and the $\sigma$-field $\mathscr{F}_{0}$ contains all $\mathbb{P}$-null sets of $\mathscr{F}$, see [47, Definition I.1.1]. According to [47, Definition I.4.1], a function $\tau:\Omega\to[0,\infty]$ is an $\mathbb{F}$-stopping time iff for every $t\in\mathbb{R}_{+}$, the set $\{\tau\leq t\}:=\{\omega\in\Omega:\tau(\omega)\leq t\}$ belongs to the $\sigma$-field $\mathcal{F}_{t}$.

Definition 2.14**.**

A stopping time $\tau$ is called accessible, see e.g. [43, section 2.1, p. 45], iff there exists an increasing sequence of stopping times $\{\tau_{n}\}_{n\in\mathbb{N}}$ with the following properties:

1. (1)

$\lim\limits_{n\to\infty}\tau_{n}=\tau$, $\mathbb{P}$-a.s., 2. (2)

for every $n$, $\tau_{n}<\tau$, $\mathbb{P}$-a.s. on $\{\tau>0\}$.

For such sequence we write $\tau_{n}\nearrow\tau$. Such a sequence $\{\tau_{n}\}_{n\in\mathbb{N}}$ will be called an announcing sequence for the accessible stopping time $\tau$.

Let us point out that in [64, Definition IV.5.4] a process which we call accessible is called predictable. On the other hand, Metivier in [47, Definition I.4.9] gives a different definition of a predictable stopping time. Fortunately, according to [47, Theorem I.6.6], $\tau$ is a predictable stopping time according to [47, Definition I.4.9] if and only if $\tau$ is an accessible according to our definition, provided the usual hypothesis satisfies, see [47, Definition I.1.1]. Let us point out that our standing assumption is that the the filtered probability space $(\Omega,\mathscr{F},\mathbb{P},\mathbb{F})$ satisfies the usual hypothesis. Therefore, in our paper, the notions of accessible and predictable stopping times are equivalent and this allows us to use later on [47, Proposition I.4.14].

For any given stopping time $\tau$, we set

[TABLE]

Note that the sets $[0,\tau)\times\Omega$ and $[0,\tau]\times\Omega$ are sometimes denoted by $[[0,\tau))$ and $[[0,\tau]]$ respectively. They are also sometimes called ”stochastic intervals”. It is useful to observe that for two stopping times $\tau_{1}$ and $\tau_{2}$ and $t>0$, the following equality holds:

[TABLE]

where $(\tau_{1}\wedge\tau_{2})(\omega):=\min\{\tau_{1}(\omega),\tau_{2}(\omega)\}$, $\omega\in\Omega$.

To prove the uniqueness of a local solution we need the following criteria of equivalent processes.

Definition 2.15**.**

*Assume that $X$ is a separable Banach space. A local $X$-valued stochastic process is a function $\xi:[0,\tau)\times\Omega\to X$, where $\tau$ is an accessible stopping time.

Suppose that $\tau$ is an accessible stopping time with an announcing sequence $\{\tau_{n}\}_{n\in\mathbb{N}}$. Then a local stochastic process $\xi:[0,\tau)\times\Omega\to X$ is called $\mathbb{F}$-progressively measurable, see e.g. [9], iff for every $n$, the stopped $X$-valued process*

[TABLE]

*is $\mathbb{F}$-progressively measurable. If the filtration $\mathbb{F}$ is unambiguous from the context, then we often skip it from using.

Two local stochastic processes $\xi_{i}:[0,\tau_{i})\times\Omega\to X$, $i=1,2$ are called equivalent, we will write $(\xi_{1},\tau_{1})\sim(\xi_{2},\tau_{2})$, if and only if $\tau_{1}=\tau_{2}$, $\mathbb{P}$-a.s. and for any $t>0$ the following holds*

[TABLE]

for almost all $\omega\in\Omega_{t}(\tau_{1})\cap\Omega_{t}(\tau_{2})$.

Assume that $p\in[1,\infty)$. By $\mathcal{M}_{loc}^{p}(\mathbb{R}_{+},E)$, we denote the space of all progressively measurable $E$-valued processes $\xi:\mathbb{R}_{+}\times\Omega\to E$ for which there exists a sequence $\{\tau_{n}\}_{n\in\mathbb{N}}$ of bounded stopping times such that $\tau_{n}\nearrow\infty,~{}\mathbb{P}$-a.s. and

[TABLE]

Assume that $p\in[1,\infty)$, $q\in[1,\infty)$, and $T>0$. By $\mathcal{M}^{p}(\mathcal{L}^{q}([0,T],E))$ we denote the space of all progressively measurable $E$-valued processes $\xi:[0,T]\times\Omega\to E$ such that

[TABLE]

The space $\mathcal{M}^{p}(\mathcal{L}^{\infty}([0,T],E))$ is defined analogously with the “norm” \bigl{(}\int_{0}^{T}\|\xi(t)\|_{E}^{q}\,dt\bigr{)}^{\frac{1}{q}} being replaced by $\mathrm{esssup}_{t\in[0,T]}\|\xi(t)\|_{E}$.

As usual, see e.g. [64, Definition IV.2.1], by $\mathbb{M}^{p}({L}^{q}([0,T],E))$ we denote the space of equivalence classes of elements of $\mathcal{M}^{p}(\mathcal{L}^{q}([0,T],E))$. Let us note that $\mathbb{M}^{p}({L}^{q}([0,T],E))$ is a closed subspace of, typically not equal to, $L^{p}(\Omega,\mathscr{F};{L}^{q}([0,T],E))$. To simplify the notation, we will often use the notation ${M}^{p}([0,T],E)$ to denote $\mathbb{M}^{p}({L}^{p}([0,T],E))$.

3. Inhomogeneous Strichartz estimates

In this section we will prove the deterministic Strichartz type estimate, see Theorem 3.2 below, which is a generalization of [41, Theorem 1.2] and is essential to tackle, both, the Dirichlet and the Neumann boundary case.

Recall that in our setting, the operator $(A,\mathscr{D}(A))$ possesses a complete orthonormal system of eigenvectors $\{e_{j}\}_{j\in\mathbb{N}}$ in $L^{2}({\mathscr{O}})$. We have denoted the corresponding eigenvalues by $\lambda_{j}^{2}$. From the functional calculus of self-adjoint operators, it is known that $\{(e_{j},\lambda_{j})\}_{j\in\mathbb{N}}$ is a sequence of the associated eigenvector and eigenvalue pair for $\sqrt{A}$. For any integer $\lambda\geq 0$, $\Pi_{\lambda}$ is defined as the spectral projection of $L^{2}({\mathscr{O}})$ onto the subspace spanned by $\{e_{j}\}_{j\in\mathbb{N}}$ for which $\lambda_{j}\in[\lambda,\lambda+1)$, i.e.

[TABLE]

At this juncture, it is relevant to note that the proof of the Strichartz estimate in deterministic setting, see e.g. [17] and [18], is based on the following estimate in the Lebesgue spaces of the spectral projector $\Pi_{\lambda}$, refer [68] for the proof.

Theorem 3.1**.**

For any smooth bounded domain ${\mathscr{O}}\subset\mathbb{R}^{2}$, the following estimate holds for all $u\in L^{2}({\mathscr{O}})$

[TABLE]

where

[TABLE]

Since the Strichartz estimates below derived, for the homogeneous and inhomogeneous wave equation, holds for both the Dirichlet and the Neumann case, from now onwards, to shorten the notation, we denote $A_{B,q}$ and $A_{B,2}$, respectively, by $A_{q}$ and $A$.

Theorem 3.2** (Deterministic Strichartz Estimates).**

Let us assume that $T>0$. Then there exists a positive constant $C_{T}$, which is increasing w.r.t. $T$ and may also depend on $p,q,r$, such that the following holds: if $u$ satisfies the following linear inhomogeneous wave equation

[TABLE]

with either boundary condition

[TABLE]

where $\nu$ is the outward normal unit vector to $\partial{\mathscr{O}}$ and $f\in L^{1}(0,T;L^{2}({\mathscr{O}}))$, then

[TABLE]

for all $(p,q,r)$ which satisfy

[TABLE]

Remark 3.3**.**

In addition to the Strichartz estiamtes (3.4) one also has the classical (including those on the velocity $u_{t}$) estimates, see [2] for the inhomogeneous part,

[TABLE]

We can assume that the constants $\bar{C}_{T}$ increase w.r.t. $T$. In a standard way, this inequality can be lifted to more regular data as follows, for any $k\geq 0$,

[TABLE]

Remark 3.4**.**

Let us observe that if for $T>0$, $C_{T}$ denotes the smallest constant for which inequality (3.4) holds for all data $u_{0},u_{1}$ and $F$ from appropriate spaces, then the function

[TABLE]

is non-decreasing (or weakly increasing as some people call). Similar results hold true for $\bar{C}_{T}$.

Remark 3.5**.**

Ibrahim and Jrad proved, see inequality (3.5) in the proof of [41, Theorem 1.2], that

[TABLE]

for $A=-\Delta_{D}$. Since the substitution of $p=q=8$ into (3.4) gives (3.8), our result generalizes [41, Theorem 1.2]. Note that in [41] the space $H^{\frac{3}{8},8}({\mathscr{O}})$ is denoted, inconsistently with current approach, by $W^{\frac{3}{8},8}({\mathscr{O}})$. Finally, let us point out that the inequality (1.8) with the Hölder space $\dot{C}^{\frac{1}{8}}({\mathscr{O}})$ in [41, Theorem 1.2] is a consequence of the Sobolev embedding, see e.g. [69, Theorem 2.8.1 (e) and Definition 1 (d) in 2.3.1].

Proof of Theorem 3.2.

Without loss of generality we assume that $T=2\pi$. The proof is divided into two cases. In the first case, we derive the Strichartz estimate for the homogeneous problem (i.e. $F=0$) and then, in second case, we prove the inhomogeneous one (i.e. $F\neq 0$) by using the homogeneous estimate from first case.

First case : Estimate for the homogeneous problem. In this case, the Duhamel formula gives

[TABLE]

where, from the functional calculus for self-adjoint operators, for each $t$, $\cos(t{\sqrt{A}})$ and $\frac{\sin(t{\sqrt{A}})}{\sqrt{A}}$ are well-defined bounded operators on $L^{2}({\mathscr{O}})$. Moreover, we have

[TABLE]

Let $\mathcal{L}_{\pm}(t)u_{0}:=e^{\pm it\sqrt{A}}u_{0}$ be the solution $u$ of $\partial_{t}u=\pm i\sqrt{A}u$ such that $u(0)=u_{0}$. In other words, \mathcal{L}_{\pm}=\bigl{(}\mathcal{L}_{\pm}(t))_{t\geq 0} is $C_{0}$-group with the generator $\pm i\sqrt{A}$. Using the Minkowski’s inequality we get

[TABLE]

Therefore, it is enough to estimate, as done in the following Steps 1-4, the $L^{p}(0,T;\mathscr{D}(A_{q}^{\frac{1-r}{2}}))$-norm of $e^{it\sqrt{A}}u_{0}$ and $\frac{\sin(t{\sqrt{A}})}{{\sqrt{A}}}u_{1}$. We will write the variables in subscript, wherever required, to avoid any confusion.

**Step 1 : ** Here we show that

[TABLE]

where $B$ is the following “modification” of ${\sqrt{A}}$ operator by considering only the integer eigenvalues, i.e.

[TABLE]

The notation $[\cdot]$ stands for the integer part and $e_{j}$ is an eigenfunction of $A$ associated to the eigenvalue $\lambda_{j}^{2}$. Before moving further we prove the boundedness property of the operator $B-\sqrt{A}$.

Lemma 3.6**.**

For every $r\in[0,1]$, the operator $B-\sqrt{A}$ is bounded on $\mathscr{D}(A^{\frac{r}{2}})$.

Proof of Lemma 3.6.

Let us fix $r\in[0,1]$. Observe that by definition of $B$ we have for every $u\in\mathscr{D}(A^{\frac{r}{2}})$,

[TABLE]

where $\{\lambda_{j}\}:=\lambda_{j}-\left[\lambda_{j}\right]$ is the fractional part of $\lambda_{j}$. Then

[TABLE]

Moreover,

[TABLE]

Hence, by the definition of norm in $\mathscr{D}(A^{\frac{r}{2}})$ we have

[TABLE]

∎

In continuation of the proof of (3.11), since $u_{0}\in L^{2}({\mathscr{O}})$, we can write

[TABLE]

By functional calculus for self-adjoint operators,

[TABLE]

where, since $[\lambda_{j}]=k$,

[TABLE]

Assume that $0<\theta<\frac{1}{2}$. Let $C:=-D_{tt}+I$ on $L^{2}(0,2\pi)$ with the periodic boundary conditions. Then, see [69], $H^{\theta,2}=\mathscr{D}(C^{\theta/2})$ with equivalent norms. The norm $\mathscr{D}(C^{\theta/2})$ is Hilbertian with the corresponding inner product

[TABLE]

We claim that the sequence $\{e^{i\cdot k}\}_{k\in\mathbb{N}}$, after normalization, is an orthonormal basis (ONB) in $\mathscr{D}(C^{\theta/2})$. Indeed, since the sequence $\{e^{i\cdot k}\}_{k\in\mathbb{N}}$, consisting of eigenvectors of $C$, is an ONB of $L^{2}(0,2\pi)$, we infer that for all $j,k\in\mathbb{N}$,

[TABLE]

Hence the claim follows.

Moreover, there exists $C_{\theta}>0$ such that

[TABLE]

Therefore, for a fixed $x\in D$, we have

[TABLE]

Hence,

[TABLE]

Therefore,

[TABLE]

Using the equivalence of the two norms we deduce that

[TABLE]

Thanks to the 1D Sobolev embedding and Lemma 2.2, we have

[TABLE]

where the space $H^{\theta,2}(0,2\pi)$, $\theta>0$, is defined by formula (2.1), with set ${\mathscr{O}}$ replaced by the interval $(0,2\pi)$. Consequently we argue as follows:

[TABLE]

Hence, by applying the last Claim with $\theta=\frac{1}{2}-\frac{1}{p}$ in view of the Parseval formula we deduce that, for fixed $x$,

[TABLE]

Combining the estimate (3.14) and (3) followed by Minkowski’s inequality and Theorem 3.1 we obtain

[TABLE]

where, from $\rho$ in Theorem 3.1, we haveIIINote that $r<\frac{3}{4}$ in the case $2\leq q\leq 8$ and $r<1$ in the complimentary case $8\leq q<\infty$.,

[TABLE]

Here it is important to highlight that, the equivalence $\|u_{0}\|_{\mathscr{D}(B^{r})}\simeq\|u_{0}\|_{\mathscr{D}(A^{r/2})}$ holds in the last step of (3), because $\mathscr{D}(A)=\mathscr{D}(B^{2})$ and the spaces $\mathscr{D}(B^{r})$ and $\mathscr{D}(A^{r/2})$, for $r\in[0,1]$, are equal to the complex interpolation spaces, between $L^{2}({\mathscr{O}})$ and, respectively, $\mathscr{D}(B^{2})$ and $\mathscr{D}(A)$, see [69, Theorem 4.3.3].

Next, since $p\geq q$, by the Minkowski inequality we obtain the following desired result

[TABLE]

which also implies that the operator $e^{itB}$ is continuous from $\mathscr{D}(A^{r/2})$ to $L_{t}^{p}(0,2\pi;L_{x}^{q}({\mathscr{O}}))$.

Step 2 : In this step we extend inequality (3.11) to operator $\mathcal{L}_{+}$, i.e. we show that

[TABLE]

Let $v(t)=e^{it{\sqrt{A}}}u_{0}$. It is clear that $v$ satisfies

[TABLE]

and, therefore, according to the Duhamel formula

[TABLE]

If we denote $e^{i(t-s)B}(-iB+i{\sqrt{A}})v(s,x)$ by $z(s,t,x)$ and $(-iB+i{\sqrt{A}})v(s,x)$ by $w(s,x)$, then using the Minkowski inequality, followed by estimate (3.11) and Lemma 3.6, we argue as follows:

[TABLE]

By putting together (3.17) and (3) we obtain

[TABLE]

Now, from the boundedness of $e^{it\sqrt{A}}$ on $\mathscr{D}(A^{r/2})$, we infer that

[TABLE]

Combining (3.20) and (3) we get

[TABLE]

Hence, again, as an application of the Minkowski inequality we get (3.16) and finish with the proof of Step 2.

Step 3: Here, by using the well-known consequence of Agmon-Douglis-Nirenberg regularity results for the elliptic operators, refer [1], we prove the required estimate of the first term in (3.10), in particular, we show

[TABLE]

We start the proof by recalling the following consequence of the Agmon-Douglis-Nirenberg regularity results for the elliptic operators. The operators

[TABLE]

and

[TABLE]

are isomorphisms. These operators will, respectively, be denoted by $A_{D,q}+I$ and $A_{N,q}+I$, or simply by $A_{q}+I$. Suppose that $u_{0}\in D(A^{k})$ for sufficiently large $k\in\mathbb{N}$ so that $Au_{0}\in\mathscr{D}(A^{r/2})$. Then, since the operators $A$ and $\mathcal{L}_{+}$ commute, we infer that for all $t\in[0,T]$,

[TABLE]

Consequently by (3.16) we get

[TABLE]

Thus, complex interpolation between (3.16) and (3.22) with $\theta=\frac{1-r}{2}$ gives the desired following estimate

[TABLE]

Hence we have completed the proof of Step 3.

Step 4: Here we incorporate the term with $u_{1}$, in (3.9), and complete the proof of the homogeneous Strichartz estimate.

Recall that $\lambda_{1}=0$ for the Neumann condition and $\lambda_{1}>0$ in the Dirichlet case. As mentioned before, we denote by $m_{0}$ the dimension of eigenspace corresponding to zero eigenvalue. It is known that $m_{0}=0$ for $A=-\Delta_{D}$ and a positive finite integer when $A=\Delta_{N}$. To proceed with the proof of this Step, as in [18], we single out the contribution of zero eigenvalue and decompose $L^{2}({\mathscr{O}})$ into the direct sum of a finite dimensional space $\ker A$ and the space orthogonal to $\ker A$, which we denote by $L^{2,+}({\mathscr{O}})$. Let us observe that if ${\mathscr{O}}$ is connected, then $\ker A$ is a one dimensional vector space consisting of constant functions. Mathematically, it means, for all $u_{1}\in L^{2}({\mathscr{O}})$,

[TABLE]

Note that the term $\Pi u_{1}$ does not exist in the Dirichlet condition. Then we argue as follows:

[TABLE]

where the last step holds due to the following argument

[TABLE]

Now, since \bigl{(}\sqrt{A}\bigr{)}^{-1} is isometry from $L^{2,+}({\mathscr{O}})$ into $\mathscr{D}(A^{\frac{1}{2}})$, by invoking (3.21) on \bigl{(}\sqrt{A}\bigr{)}^{-1} $((\mathds{1}-\Pi)u_{1})$ we get

[TABLE]

We mention that all the computations we have done so far in Steps 1-4 would work if we replace $\mathcal{L}_{+}$ by $\mathcal{L}_{-}$. Combining (3.23) and (3) we obtain

[TABLE]

This finishes the proof of Step 4 and, in particular, the first case.

Second case: when $L^{1}(0,2\pi;L^{2}({\mathscr{O}}))\ni F\neq 0$: Due to the Duhamel formula

[TABLE]

Applying the first case and using the calculation of (3) and (3) we get

[TABLE]

Hence we have proved the Theorem 3.2. ∎

4. Stochastic Strichartz estimates

This section is devoted to prove a stochastic Strichartz inequality, which is sufficient to apply the Banach Fixed Point Theorem in the proof of a local well-posedness result for problem (1.2), see Theorem 6.1 in Section 6.

4.1. Main assumptions

Here we describe the main assumptions we consider in Sections 4 – 6. Let us set

[TABLE]

where $q\in[2,\infty)$, $r\in[0,1]$ and $p=p(q,r)$ satisfies the equality (3.5). Let us define the following Banach space. For fixed $T>0$, we put

[TABLE]

Obviously, all these four spaces are (separable) Banach spaces with naturally defined norms, i.e.

[TABLE]

By $\mathbb{M}^{p}(Z_{T})$ we denote the Banach space of (equivalence classes) of all $E$-valued progressively measurable processes $\{u(t),t\in[0,T]\}$ having a continuous $H_{A}$-valued modification and satisfying

[TABLE]

4.2. Martingales

In order to define the Itô type integrals for a Banach space valued stochastic process, we restrict ourself to, the so called, $M$-type 2 Banach spaces which are defined as follows.

Definition 4.1**.**

A Banach space $E$ is of M-type $2$ iff there exists a constant $L:=L_{2}(E)>0$ such that for any $E$-valued martingale $\{M(n),\;n\in\mathbb{N}\}$ the following holds:

[TABLE]

where, as usual, we set $M(-1)=0$.

For an interval $I\subseteq\mathbb{R}$, we say that an $E$-valued process $\{M(t),t\in I\}$ is an $E$-valued martingale iff $M(t)\in L^{1}(\Omega,\mathscr{F},\mathbb{P};E)$ for $t\in I$ and

[TABLE]

4.3. Burkholder inequality

To prove the main result of this section we need the following consequence of the Kahane-Khintchin inequality and the Itô-Nisio Theorem, see [38]. For any $\Lambda\in\gamma(K,E)$, by the Itô-Nisio Theorem, the series $\sum_{j=1}^{\infty}\beta_{j}\Lambda(f_{j})$ is $\mathbb{P}$-a.s. convergent in $E$, where $\{f_{j}\}_{j\in\mathbb{N}}$ and $\{\beta_{j}\}_{j\in\mathbb{N}}$ are as in Definition 2.3. Then, as an application of the Kahane-Khintchin inequality, X. Fernique [34] proved that, for any $p\in[1,\infty)$, there exists a positive constant $C(p,E)$ such that,

[TABLE]

This inequality tells that the convergence in $L^{2}(\Omega,\mathscr{F},\mathbb{P};E)$ can be replaced by a condition of convergence in $L^{p}(\Omega,\mathscr{F},\mathbb{P};E)$ for some (or any) $p\in[1,\infty)$. Furthermore, we need the following version of the Burkholder inequality which holds in our setting, refer [53] for the proof.

Theorem 4.2** (Burkholder inequality).**

Let $E$ be a $M$-type 2 Banach space and $p\in[1,\infty)$. If $\xi\in\mathcal{M}_{loc}^{2}(\mathbb{R}_{+},\gamma(K,E))$, then the following conditions hold.

• (i)

There exists an $E$-valued continuous process $\{\Xi(t)$, $t\in[0,T]\}$, such that

[TABLE]

The random variable $\Xi(t)$ will be denoted, unless a danger of ambiguity, by $\int_{0}^{t}\xi(s)\,dW(s)$.

• (ii)

There exists a constant $B_{p}(E)>0$, independent of $\xi$, such that for each stopping time $\tau>0$,

[TABLE]

Finally, (4.12) holds provided only that $\xi=\{\xi(t),0\leq t<\tau\}$ is a local progressively measurable process.

Remark 4.3**.**

[Warning!] The Itô integral $\int_{0}^{t}\xi(s)\,dW(s)$ is, by definition, an element of $L^{p}(\Omega,\mathscr{F}_{t};E)$, thus an equivalence class of a certain class of $\mathscr{F}_{t}$/$\mathscr{B}(E)$-measurable functions. In what follows we will use a formally imprecise formulation as $\Xi(t)=\int_{0}^{t}\xi(s)\,dW(s)$ $\mathbb{P}$-a.s. used in (4.11), instead of the correct, but awkward, one that $\Xi(t)$ belongs to $\int_{0}^{t}\xi(s)\,dW(s)$.

We ask the reader to refer [11, Corollary 3.7] for a proof of the following result. It is important to mention here that the range of $p$ assumed in the statement of [11, Corollary 3.7] is incorrect as from the proof in [11] it is clear that the result only holds true if $p\geq 2$.

Corollary 4.4**.**

Let $E$ be a M-type 2 Banach space and $p\in[2,\infty)$. Then there exists a constant $\hat{B}_{p}(E)$, depending on $E$, such that for every $T\in(0,\infty]$ and every $L^{p}(0,T;E)$-valued progressively measurable process $\{\zeta(s),s\in[0,T)\}$,

[TABLE]

For a $\gamma(K,H)$-valued random variable $\xi$, let us define a $\gamma(K,L^{p}(0,T;E))$-valued process $\Xi=\{\Xi_{r}:r\in[0,T]\}$ by

[TABLE]

We will need the following auxiliary result.

Lemma 4.5**.**

Assume that $T>0$. Then there exists a positive constant $C(p,T,H)$ such that for every $\gamma(K,H)$-valued progressively measurable process $\xi$, the $\gamma(K,L^{p}(0,T;E))$-valued process $\{\Xi_{r}:r\in[0,T]\}$, defined by formula (4.14), is progressively measurable and,

[TABLE]

Proof of Lemma 4.5.

Let us consider a sequence $\{\beta_{j}\}_{j\in\mathbb{N}}$ of i.i.d. $N(0,1)$ random variables on probability space $(\Omega,\mathscr{F},\mathbb{P})$, and an orthonormal basis $\{f_{j}\}_{j\in\mathbb{N}}$ of the separable Hilbert space $K$. We first observe that the random variable $\Xi_{r}$ is well-defined because by Theorem 3.2, for each $r\in[0,T]$ and $x\in H$, the solution of the following homogeneous wave equation

[TABLE]

belongs to $L^{p}(r,r+T;E)$. In particular,

[TABLE]

and the map

[TABLE]

is linear and continuous. Moreover, we have $\sup_{r\in[0,T]}\|\Lambda_{r}\|<\infty$. By the above argument and (4.14), we infer that

[TABLE]

Next, for each $r\in[0,T]$ and $x\in H$, the deterministic Strichartz estimate (3.4) yields

[TABLE]

with the RHS being independent of $r$. Consequently, we have

[TABLE]

To move further with the proof let us set the following useful notation:

[TABLE]

We have the following result.

Lemma 4.6**.**

If $x\in H$, then the function

[TABLE]

is continuous.

Proof of Lemma 4.6.

We first assume that $x\in\mathscr{D}(A^{k-\frac{1}{2}})$, where $k\in\mathbb{N}$ is such that $\mathscr{D}(A^{k})\hookrightarrow E$. Since $\mathscr{D}(A^{k-\frac{1}{2}})$ is dense in $H$, it is sufficient to prove the lemma for $x\in\mathscr{D}(A^{k-\frac{1}{2}})$. Note that, by property (3.7) for $x\in\mathscr{D}(A^{k-\frac{1}{2}})$ we have

[TABLE]

Since $\|f_{r}-f_{r_{0}}\|_{L^{p}(0,T;\mathbb{R})}^{p}=r-r_{0}$, we infer that the function

[TABLE]

is continuous. Next we claim that if $g(\cdot,x)\in\mathrm{C}([0,T];E)$, then the map

[TABLE]

is continuous. For this we note that for $0\leq r<s\leq T$ we have

[TABLE]

because

[TABLE]

Concerning the first integral on the RHS of (4.20) we have

[TABLE]

For the second integral in the RHS of (4.20), we have

[TABLE]

where, by the uniform continuity of function $g(\cdot,x):[0,T]\to E$,

[TABLE]

Hence the continuity of function $h$ follows and we are done with the proof of Lemma  4.6. ∎

Thus, by coupling the Lemma 4.6 with Proposition 2.10 and [38, Corollary 1.1.29], we infer that the process $\Xi$ is progressively measurable.

It only remains to prove the inequality (4.15). For this aim let us fix $r\in[0,T]$. By invoking the inhomogeneous Strichartz estimate from Theorem 3.2 and (4.10), followed by (4.18), we obtain

[TABLE]

Hence the proof of Lemma 4.5 is complete. ∎

The following main result of this section is one of the most important ingredient in the proof of the local existence theorem in Section 6.

Theorem 4.7** (Stochastic Strichartz Estimates).**

Let us assume that $T>0$ and $p\in[2,\infty)$. Then there exist constants111The constant $K$ depends on $T$ only in the case of Neumann boundary conditions. $\tilde{K}_{T}:=K(p,T,H)>0$ and $\tilde{C}_{T}:=\tilde{C}(p,T,E,H)>0$ such that if a process $\xi$ belongs to $\mathcal{M}^{p}(\mathcal{L}^{2}([0,T],\gamma(K,H))$, then the following assertions hold.

• (I)

There exists an $\mathbb{F}$-adapted, and $H_{A}\times H$-valued222Let us recall that $H_{A}={\mathscr{D}(A^{\frac{1}{2}})}$. * continuous process $(\tilde{u},\tilde{v})$ such that*

[TABLE]

[TABLE]

[TABLE]

• (II)

There exists an $E$-valued progressively measurable process $\tilde{\tilde{u}}$ such that

[TABLE]

[TABLE]

*where $i:H_{A}\hookrightarrow H$ and $j:E\hookrightarrow H$ are the natural embeddings. *

• (III)

Moreover, if processes $\xi_{1}$ and $\xi_{2}$ are equivalent, then so are the corresponding processes $\tilde{\tilde{u_{1}}}$ and $\tilde{\tilde{u_{2}}}$. In particular, the map

[TABLE]

extends in a unique way to the following bounded and linear map

[TABLE]

Remark 4.8**.**

*Suppose that $\tau$ is a stopping time such that $\tau\leq T$ for some $T>0$ and $\xi(s)$, $s\in[0,\tau)$ is an $\gamma(K,E)$-valued progressively measurable process. Since the process $\mathds{1}_{[0,\tau)}(s)$, $s\in[0,\infty)$ is well-measurable, see [47, Proposition 4.2] and, see [47, Theorem 1.6], the $\sigma$-field of well-measurable sets is smaller than the $\sigma$-field of progressively measurable sets, it follows that the process $\mathds{1}_{[0,\tau)}(s)$, $s\in[0,\infty)$ is progressively measurable. In particular, the process $\{\mathds{1}_{[0,\tau)}(s)\xi(s),s\in[0,\infty)\}$ is progressively measurable. Hence, by applying inequalities (4.21) and (4.24) to the process $\mathds{1}_{[0,\tau)}\xi$ we infer the following stopped versions of those inequalities.

• (i)

There exists an $\mathbb{F}$-adapted, $H_{A}\times H$-valued continuous local process (\tilde{u},\tilde{v})=\bigl{(}(\tilde{u}(t),\tilde{v}(t)),t\in[0,T]\bigr{)}, such that

[TABLE]

[TABLE]

[TABLE]

• (ii)

There exists an $E$-valued progressively measurable local process $\tilde{\tilde{u}}(t)$, $t\in[0,\tau]$, such that

[TABLE]

[TABLE]

Remark 4.9**.**

It follows from the proof that $\tilde{K}_{T}\leq M_{1}e^{mT}B_{p}(\mathcal{H})$ for some constants $m\geq 0$ and $M_{1}\geq 1$ and $\tilde{C}_{T}\leq C_{T}~{}C(p,H)~{}\hat{B}_{p}(E)$.

Proof of part (I) of Theorem 4.7.

In what follows we fix either the Dirichlet or the Neumann boundary conditions. Let us fix $p\in[2,\infty)$. To prove the first assertion, let us consider the Hilbert space $\mathcal{H}:=H_{A}\times H=\mathscr{D}(A^{\frac{1}{2}})\times H$ and a linear operator $\mathcal{A}$ on $\mathcal{H}$ defined by

[TABLE]

It is well known that since $A$ is non-negative and self-adjoint in $L^{2}({\mathscr{O}})$, one may prove that $\mathcal{A}$ generates a $C_{0}$-group on $\mathcal{H}$, denoted by $\{S(t)\}_{t\geq 0}$. Moreover, for $t\geq 0$,

[TABLE]

i.e., using a matrix notation,

[TABLE]

It follows, with being $\pi_{1}:\mathcal{H}\to H_{A}$ the natural projection, that for $t\geq 0$,

[TABLE]

Let us introduce the following auxiliary $\gamma(K,\mathcal{H})$-valued process $\tilde{\xi}$

[TABLE]

Our argument now is based on [37]. We begin by observing, see e.g. [73, Theorem 12.2], that there exists an $\mathcal{H}$-valued continuous process $\tilde{\mathfrak{q}}=\{\tilde{\mathfrak{q}}(t),t\in[0,T]\}$ such that

[TABLE]

Since $\{S(t)\}_{t\geq 0}$ is a $C_{0}$-group, we infer that the process $\tilde{\mathfrak{u}}$ defined by

[TABLE]

is a continuous $\mathcal{H}$-valued and that

[TABLE]

With $\pi_{2}:\mathcal{H}\to H$ being the natural projection, we define continuous $H_{A}$ and $H$-valued processes $\tilde{u}$ and $\tilde{v}$, respectively, by

[TABLE]

Applying identity (4.34) to the two previous equalities (4.36)-(4.37) we infer that

[TABLE]

Thus, we proved that $\tilde{u}$ is ${H}_{A}$-valued continuous and $\mathbb{F}$-adapted process satisfying equality (4.22). Moreover, using the Burkholder inequality (4.12) and the bound property of $C_{0}$-group, we get the following train of inequalities:

[TABLE]

where $K_{T}\leq M_{1}e^{mT}$ for some constants $m\geq 0$ and $M_{1}\geq 1$. This yields inequality (4.21) and, in particular, assertion of part (i).

Equality (4.29) follows from Proposition C.1 and equalities (4.37). ∎

Proof of part (II) of Theorem 4.7.

We split the proof into two steps. First we will prove the theorem for more regular processes. Then we will transfer the results to the right class of processes by employing a suitable approximation.

Step 1: We begin by observing that by the classical Sobolev embedding theorem there exists natural number $k$ such that the Hilbert space

[TABLE]

Let us fix $p\in[2,\infty)$. Let us assume that a process $\xi$ belongs to $\mathcal{M}^{p}(\mathcal{L}^{2}([0,T],\gamma(K,\mathscr{D}(A^{k})))$. By assertion (i), we infer that there exists an $\mathbb{F}$-adapted ${\mathscr{D}(A^{k+\frac{1}{2}})}$-valued continuous process $\tilde{u}$ which satisfies condition (4.22) and the following inequality

[TABLE]

Also, let us note that, in view of our additional assumption (4.38), the process $\tilde{u}$ is an $\mathbb{F}$-adapted $H_{A}$-valued continuous (hence progressively measurable), and

[TABLE]

Next, we define an $\gamma(K,L^{p}(0,T;\mathscr{D}(A^{k})))$-valued process $\Xi=\{\Xi_{r}:r\in[0,T]\}$ by formula (4.14). This process, in view of Lemma 4.5, is progressively measurable and, by the Burkholder inequality (4.13) together with inequality (4.15), it satisfies the following inequality

[TABLE]

Let us choose an $\mathscr{F}_{T}$/$\mathscr{B}(L^{p}(0,T;E))$-measurable function $\tilde{\tilde{\mathbf{u}}}:\Omega\to L^{p}(0,T;E)$ such that, see Remark 4.3,

[TABLE]

By the first part of Proposition B.1, there exists an $\mathscr{B}([0,T])\otimes\mathscr{F}_{T}$/$\mathscr{B}(E)$-measurable function

[TABLE]

and a set $\Omega^{\prime}\subset\Omega$ such that $\mathbb{P}(\Omega^{\prime})=1$ and for every $\omega\in\Omega^{\prime}$,

[TABLE]

Later on we will show that $\tilde{\tilde{u}}$ is progressively measurable process. Then, by inequality (4.39) we infer that process $\tilde{\tilde{u}}$ satisfies inequality (4.24).

Let us recall that $i:H_{A}\hookrightarrow H$ and $j:E\hookrightarrow H$ are the natural embeddings. We define corresponding Nemytski type embeddings $I$ and $J$ by

[TABLE]

and observe that both $I$ and $J$ are continuous. Therefore we deduce that $J\circ\tilde{\tilde{\mathbf{u}}}$ is an $L^{2}(0,T;H)$-valued random variable and

[TABLE]

Since the process $\tilde{u}$ has $\mathbb{P}$-almost surely continuous $H_{A}$-valued trajectories, by [31, Proposition 3.18] it induces, in a natural way, an $\mathscr{F}_{T}$/$\mathscr{B}(\mathrm{C}([0,T];H_{A}))$-measurable function $\tilde{\mathbf{u}}:\Omega\to\mathrm{C}([0,T];H_{A})$. Because the map $I:\mathrm{C}([0,T];H_{A})\to L^{2}(0,T;H)$ is continuous, $I(\tilde{\mathbf{u}}):\Omega\to L^{2}(0,T;H)$ is $\mathscr{F}_{T}$/$\mathscr{B}(L^{2}(0,T;H))$-measurable. We claim that

[TABLE]

Proof of equality (4.43).

Since $H$ is a separable Hilbert space, the Banach space $\mathrm{C}([0,T];H)$ is also separable. Let us choose a dense subset $\{h_{n}:n\in\mathbb{N}\}$ in $\mathrm{C}([0,T];H)$. Let us choose and fix $n\in\mathbb{N}$ and define a linear and bounded operator

[TABLE]

Thus, we infer that

[TABLE]

and

[TABLE]

Let us note that by definitions (4.44) of $H_{n}$, (4.42) of $J$ and (4.14) of $\Xi_{r}$ we have, for all $k\in K$ and $r\in[0,T]$, the following equality

[TABLE]

Therefore, we infer that $\mathbb{P}$-almost surely

[TABLE]

where the last equality is a consequence of the stochastic Fubini Theorem [16], which is a generalization of [22, Theorem 2.4.16] and [70, Theorem 2.2].

On the other hand, by the definition (4.41) of the map $I$, we have

[TABLE]

Thus, from (4.45), (4.47) and (4.46), we infer that for every $n\in\mathbb{N}$,

[TABLE]

By the density of the countable set $\{h_{n}\}$ in $L^{2}(0,T;H)$ we deduce (4.43). ∎

From the just proven equation (4.43) and equality (4.40) we infer that

[TABLE]

Hence, by the second part of Proposition B.1, we infer that the $H$-valued processes $i(\tilde{u}):[0,T]\times\Omega\to H$ and $j(\tilde{\tilde{u}}):[0,T]\times\Omega\to H$, are $\mathrm{Leb}\otimes\mathbb{P}$ equal. Since, the former is $H$-valued progressively measurable, by the Kuratowski Theorem, see e.g. [59, Corollary I.3.3] and the argument in the proof of [8, Proposition A.1] we infer that process $\tilde{\tilde{u}}$ is $E$-valued progressively measurable. This concludes the proof of Step 1.

Step 2: The result follows by applying Step 1. Let $\xi$ be a progressively measurable process from the space $\mathcal{M}^{p}(\mathcal{L}^{2}([0,T],\gamma(K,H))$, and $k\in\mathbb{N}$ as in Step 1.

We choose a sequence $\{\xi_{n}\}_{n\in\mathbb{N}}$ of processes from $\mathcal{M}^{p}(\mathcal{L}^{2}([0,T],\gamma(K,\mathscr{D}(A^{k})))$ s.t.

[TABLE]

We denote the corresponding processes for $\xi_{n}$, from the previous step, by $\tilde{u}_{n}$ and $\tilde{\tilde{u}}_{n}$. By Step 1, for each $n$, the processes $\tilde{u}_{n}$ and $\tilde{\tilde{u}}_{n}$ satisfy the condition (4.25), the process $\tilde{u}_{n}$ satisfies inequality (4.21) and the process $\tilde{\tilde{u}}_{n}$ satisfies inequality (4.24). Thus, both sequences are Cauchy in the appropriate Banach spaces $\mathbb{M}^{p}({L}^{\infty}([0,T],H_{A}))$ and $M^{p}([0,T],E)$, respectively. Hence, there exist unique elements in those spaces, whose representatives, respectively, we denote by $\tilde{u}$ and $\tilde{\tilde{u}}$. Because the convergence (4.48) is sufficiently fast, we deduce that $\mathbb{P}$-almost surely, $\tilde{\tilde{u}}_{n}\to\tilde{\tilde{u}}$ in $L^{p}(0,T;E)$ and $\tilde{u}_{n}\to\tilde{u}$ in $\mathrm{C}([0,T];H_{A})$. Hence, we infer that $\tilde{u}$ is $H_{A}$-valued $\mathbb{F}$-adapted and continuous process and $\tilde{\tilde{u}}$ is an $E$-valued progressively measurable process. Moreover, the processes $\tilde{u}$ and $\tilde{\tilde{u}}$ satisfy the condition (4.25). Hence we are done with the proof of part (ii) of Theorem 4.7.

∎

Proof of part (III) of Theorem 4.7.

This part follows straightforwardly from the second part of Proposition B.1. ∎

5. Local well-posedness - preliminary results

The aim of this section is to formulate and prove some preliminary results which will be helpful in Section 6 where we show the existence and uniqueness of solutions to the stochastic wave equation (1.2). Recall that we are working in the setting mentioned in the subsection 4.1.

Let us also recall the following definitions. For any $T>0$, we put

[TABLE]

Obviously, $X_{T}$, $Y_{T}$, $Q_{T}$ and $Z_{T}$ are (separable) Banach spaces with naturally defined norms, see e.g. (4.6)-(4.9).

By $\mathbb{M}^{p}(Z_{T})$ we denote the Banach space of (equivalence classes) of all $E$-valued progressively measurable processes $\{u(t),t\in[0,T]\}$ having a continuous $H_{A}$-valued modification and satisfying

[TABLE]

If $\tau$ is a bounded stopping time, by $\mathbb{M}^{p}(Z_{\tau})$ we mean the Banach space of (equivalence classes of) all progressively measurable processes $\xi:[0,\tau]\times\Omega\to E$ which have a continuous $H_{A}$-valued modification such that for each $\omega\in\Omega$, $\xi(\cdot,\omega)\in Z_{\tau(\omega)}$ and

[TABLE]

5.1. SNLWE and assumptions

Here we recall the stochastic nonlinear wave equation we consider here and state the assumptions on the drift and diffusion terms. To be precise, we consider the following Cauchy problem for stochastic nonlinear wave equation with the Dirichlet or the Neumann boundary condition

[TABLE]

where $A$ is either $-\Delta_{D}$ or $-\Delta_{N}$; $(u_{0},u_{1})\in H_{A}\times L^{2}({\mathscr{O}})$ and $W=\{W(t),t\geq 0\}$ is a cylindrical Wiener process on some real separable Hilbert space $K$ such that $K\hookrightarrow L^{\infty}({\mathscr{O}})$ and for some orthonormal basis $\{f_{j}\}_{j\in\mathbb{N}}$ of $K$,

[TABLE]

We assume the following hypotheses for the nonlinearity $F$ and the diffusion coefficient $G$ in (5.3).

1. A.1

Assume that a map

[TABLE]

is such that $F(0)=0$ and there exists a $\gamma_{F}>0$ such that for every $M\in(0,1)$ there exists a positive real number $C_{F}:=C(M,\gamma_{F})$, such that the following holds

[TABLE]

provided

[TABLE] 2. A.2

Assume that a map

[TABLE]

is such that $G(0)=0$ and there exists a $\gamma_{G}>0$ such that for every $M\in(0,1)$ there exists a positive real number $C_{G}:=C(M,\gamma_{G})$ such that if $u,v$ satisfy (5.6), then

[TABLE]

Remark 5.1**.**

Without loss of generality we will assume that $\gamma_{F}=\gamma_{G}=:\gamma$.

Remark 5.2**.**

Note that, for given $M<1$, since $F$ and $G$ are generalized Lipschitz functions, i.e. satisfy (5.5) and (5.7), respectively, on an open ball $B(0,M)$ of radius $M$ around [math] in space $H_{A}$, by Theorem E.3, there exist maps $\tilde{F}$ and $\tilde{G}$, defined on $H_{A}\cap E$, taking values in spaces $H$ and $\gamma(K,H)$, respectively, such that the inequalities (5.5) and (5.7) hold true for every $u,v\in H_{A}\cap E$. In particular, with Remark 5.1, there exists a $\gamma>0$ such that for every $u,v\in H_{A}\cap E$

[TABLE]

The next two lemmata are a straightforward but important consequences of Remark 5.2 and Assumptions A.1 and A.2.

Lemma 5.3**.**

Let us assume that the function $F:H_{A}\cap E\to H$ satisfies Assumption A.1 with $\gamma=\gamma_{F}$ and let $T>0$ and $p\geq\gamma$. Let $\tilde{F}$ be the extension of $F$ as mentioned above in Remark 5.2. If $M\in(0,1)$ and $C_{\tilde{F}}:=3C(M,\gamma)>0$ are as in Remark 5.2, then there exists $K(\gamma)>0$ such that the following inequality holds

[TABLE]

provided $u_{1},u_{2}\in Z_{T}$.

Remark 5.4**.**

Lemma 5.3 means that the function $\tilde{F}:X_{T}\to L^{1}(0,T;H)$ is uniformly Lipschitz on the sets $\{u\in X_{T}:\|u\|_{Y_{T}}\leq R\}$, $R>0$. Analogous remark can be made for Lemma 5.5.

Proof of Lemma 5.3.

Let us choose and fix $u_{1},u_{2}\in Y_{T}\cap X_{T}=Z_{T}$. Then, by using inequality (5.8), followed by the Hölder inequality, we get

[TABLE]

Hence Lemma 5.3 follows. ∎

Lemma 5.5**.**

Let us assume that the function $G:H_{A}\cap E\to\gamma(K,H)$ satisfies Assumption A.2 with $\gamma=\gamma_{G}$ and let $T>0$ and $p\geq 2\gamma$. Let $\tilde{G}$ be the extension of $G$ as in Remark 5.2. If $M\in(0,1)$ and $C_{\tilde{G}}:=3C(M,\gamma)>0$ are as in Remark 5.2, then there exists $K(\gamma)>0$ such that the following inequality holds

[TABLE]

provided $u_{1}$ and $u_{2}$ belong to $Z_{T}$.

Proof of Lemma 5.5.

Let us choose and fix $u_{1},u_{2}\in Y_{T}\cap X_{T}$. Then, invoking inequality 5.9 and the Hölder inequality, we obtain

[TABLE]

Hence the proof of Lemma 5.5 is complete. ∎

To prove the main result of this Section 5 we need the following known results. The first one is from [66].

Theorem 5.6**.**

*[Moser-Trudinger Inequality]

Let ${\mathscr{O}}\subseteq\mathbb{R}^{2}$ be a domain (bounded or unbounded) and $\alpha\leq 4\pi$. Then*

[TABLE]

Moreover, this result is sharp in the sense that if $\alpha>4\pi$ then $C(\alpha,{\mathscr{O}})=\infty$.

The next required result is the well-known Logarithmic inequality from [58].

Theorem 5.7**.**

Let ${\mathscr{O}}$ be a domain in $\mathbb{R}^{d}$. Let $p,q,m\in\mathbb{R}$ satisfy $1<p<\infty,1\leq q<\infty$, and $m>\frac{d}{q}$. Then there exists a positive constant $L$ such that for all $u\in H^{\frac{d}{p},p}({\mathscr{O}})\cap H^{m,q}({\mathscr{O}})$ the following holds,

[TABLE]

In the next two results we provide an example of functions $f$ and $g$ such that the corresponding $F$ and $G$ satisfy the assumptions A.1 and A.2, respectively. The example below comes from [40] and [41] when $E$ is a suitable Hölder space. We will prove the next result in detail because we need a slightly more general version of the Moser-Trudinger inequality and the Logarithmic estimate, respectively, see Theorem 5.6 and 5.7, than those used in [40] and [41].

Lemma 5.8**.**

Assume that ${\mathscr{O}}\subseteq\mathbb{R}^{2}$ is a bounded domain. Let $L$ be a constant from the Theorem 5.7. Let $h:\mathbb{R}\to\mathbb{R}$ be a function defined by h(x)=\pm x\bigl{(}e^{4\pi x^{2}}-1\bigr{)} for $x\in\mathbb{R}$. Assume that a pair $(q,r)$ of positive numbers satisfies

[TABLE]

Then for every $M\in(0,1)$ there exists a constant $C_{M,L}>0$, which depending only on $M$ and $L$, such that

[TABLE]

provided $u,v$ satisfy the following condition

[TABLE]

In the next result, which is about a generalized Nemytskii operator $G$ associated with function $g$, $\{f_{j}\}_{j\in\mathbb{N}}$ is an ONB of a Hilbert space $K$.

Lemma 5.9**.**

Assume that condition (5.4) holds. Assume that g(x)=x\bigl{(}e^{4\pi x^{2}}-1\bigr{)}, $x\in\mathbb{R}$ and a pair $(q,r)$ exists which satisfies condition (5.12). Let $L$ is a constant from the Theorem 5.7 and $G$ be defined by

[TABLE]

Then for every $M\in(0,1)$ the following inequality holds

[TABLE]

where

[TABLE]

for all $u,v\in H^{1,2}({\mathscr{O}})\cap H^{1-r,q}({\mathscr{O}})$ satisfying condition (5.14).

Remark 5.10**.**

Both Lemmata 5.8 and 5.5 are applicable to spaces defined in (4.1) because

[TABLE]

Proof of Lemma 5.5.

Let $u$ and $v$ belong to $H^{1,2}({\mathscr{O}})\cap H^{1-r,q}({\mathscr{O}})$ satisfying condition (5.14). By assumption (5.4) and Lemma 5.8 (applied to $h=g$) we infer that

[TABLE]

as required. Hence the result follows by applying the inequality (5.13). ∎

Proof of Lemma 5.8.

We only prove the result for $h(x)=x(e^{4\pi x^{2}}-1)$, since the proof for the function $-x(e^{4\pi x^{2}}-1)$ is analogous. We begin here with the following observation which is a consequence of the Mean Value Theorem. If $u,v\in\mathbb{R}$, then the following equality holds

[TABLE]

Let us now fix $M\in(0,1)$ and choose $\zeta\in(0,1)$ and $\varepsilon>0$, such that

[TABLE]

Let us take arbitrary $u,v\in H^{1,2}({\mathscr{O}})\cap H^{1-r,q}({\mathscr{O}})$ satisfying condition (5.14). Applying the above for $u=u(x)$ and $v=v(x)$, for a fixed $x\in\mathscr{O}$ we get, with $u_{\theta}(x):=(1-\theta)u(x)+\theta v(x)$,

[TABLE]

Thus, we infer that

[TABLE]

Applying the Minkowski inequality gives

[TABLE]

Then due to the Hölder inequality, the Sobolev embedding and the following basic inequality

[TABLE]

we infer that

[TABLE]

Moreover, since $\theta\in(0,1)$, $\|u_{\theta}\|_{H^{1,2}({\mathscr{O}})}\leq M$ and therefore (5.15) holds. Thus, the Moser-Trudinger inequality from Theorem 5.6 gives

[TABLE]

Invoking the estimate from Theorem 5.7, which is possible due to (5.12) and Lemma 2.2, we obtain

[TABLE]

Using the fact that if $B_{1}>1,B_{2}>0$, then the function x\mapsto x^{2}\Bigl{(}1+\log\bigl{(}B_{1}+\frac{B_{2}}{x}\bigr{)}\Bigr{)} is non-decreasing, we deduce that,

[TABLE]

Let us put

[TABLE]

Note that, since $(1+\varepsilon)M^{2}<1$, from (5.15) we get that $\gamma<2\pi L^{2}$. Next, from (5.1), (5.1), (5.18) and (5.19), we infer that

[TABLE]

Hence Lemma 5.8 follows. ∎

Remark 5.11**.**

For polynomial functions a stronger version of previous two lemmata hold, see e.g. [11].

5.2. Definition of a local mild solution

In this subsection we introduce the definitions of local and maximal local solutions we adopt in this paper. They are modifications of definitions used in earlier papers, see e.g. [10].

Definition 5.12**.**

Assume that a triple $(p,q,r)$ satisfies condition (3.5). Assume further that $(u_{0},u_{1})\in H_{A}\times H$ satisfies the following condition

[TABLE]

• A.

A local mild solution to problem (5.3) is a $\mathscr{D}(A^{\frac{1}{2}})$-valued continuous and $\mathbb{F}$-adapted process $u=\{u(t):t\in[0,\tau)\}$ satisfying the following conditions

1. (1)

$\tau$* is an accessible stopping time,* 2. (2)

the condition (5.21) is preserved, i.e.

[TABLE] 3. (3)

there exists an announcing sequence $\{\tau_{k}\}_{k\geq 1}$ of the stopping times for $\tau$, such that

[TABLE]

and, for all $t\geq 0$ and $k\in\mathbb{N}$,

[TABLE]

where $I_{\tau_{k}}(G)$ is a process defined by

[TABLE]

• B.

A local mild solution $u=\{u(t):t\in[0,\tau)\}$ to problem (5.3) is unique iff for any other local solution $\hat{u}=\{\hat{u}(t):t\in[0,\hat{\tau})\}$ to problem (5.3), the restricted processes {\left.\kern-1.2ptu\vphantom{\big{|}}\right|_{[0,\tau\wedge\hat{\tau})\times\Omega}} and {\left.\kern-1.2pt\hat{u}\vphantom{\big{|}}\right|_{[0,\tau\wedge\hat{\tau})\times\Omega}} are equivalent.

• C.

A local mild solution $u=\{u(t):t\in[0,\tau)\}$ to problem (5.3) is not maximal iff there exists a local solution $\hat{u}=\{\hat{u}(t):t\in[0,\hat{\tau})\}$ to problem (5.3) such that $\mathbb{P}(\tau\leq\hat{\tau})=1$, $\mathbb{P}(\tau<\hat{\tau})>0$ and processes $u$ and {\left.\kern-1.2pt\hat{u}\vphantom{\big{|}}\right|_{[0,\tau)\times\Omega}} are equivalent. Otherwise, a local mild solution is called maximal.

Remark 5.13**.**

The definition of the process $I_{\tau_{n}}(G)$ is explained in Lemma A.1 of Appendix A. The use of processes $I_{\tau_{n}}(G)$ was first introduced for the SPDEs of parabolic type in [7] and [22] and in [10] for the hyperbolic SPDEs. The definition we use above is only in terms of the process $u$ and thus it is different from the one used in [10] which is in terms of pair processes $(u,u_{t})$. In Appendix C we discuss the equivalence between these two approaches.

It can be shown that the concept of a local maximal solution introduced in part C. of Definition 5.12 is equivalent to the following set theoretical one. A natural continuation of this new definition is the so called “Amalgamation Lemma”, see Lemma 6.15, which is a generalization of [33, Lemma III 6A and 6B]. See also Definition 3.11 in [9].

Definition 5.14**.**

Let us denote the set of all local solutions $(u,\tau)$ to the problem (5.3) by $\mathcal{LS}$. For any two elements $(u,\tau),(v,\sigma)\in\mathcal{LS}$ we write that $(u,\tau)\preceq(v,\sigma)$ iff $\tau\leq\sigma$ $\mathbb{P}$-a.s. and $v_{|[0,\tau)\times\Omega}\sim u$, see Definition 2.15 for the notation $\sim$. We write $(u,\tau)\prec(v,\sigma)$ iff $(u,\tau)\preceq(v,\sigma)$ and $(u,\tau)\not\sim(v,\sigma)$. It is straightforward to show that $\preceq$ is a partial order on $\mathcal{LS}$.

We say that $(u,\tau)$ is a maximal element of $(\mathcal{LS},\preceq)$ iff there is no $(v,\sigma)\in(\mathcal{LS},\preceq)$ such that $(u,\tau)\prec(v,\sigma)$. Each maximal element $(u,\tau)$ in the set $(\mathcal{LS},\preceq)$ is called a maximal local solution to the problem (5.3).

6. The main results

6.1. Statements of the results on the existence and the uniqueness of solutions

The main results of the present paper, i.e. the existence of an unique local maximal solution to the problem (5.3), will be proved in this subsection.

Theorem 6.1**.**

Let us assume that a triple $(p,q,r)$ satisfies condition (3.5). Let $H,H_{A}$ and $E$ be Hilbert and respectively Banach spaces defined in (4.1). Let us assume that the maps $F:E\cap H_{A}\to H$ and $G:E\cap H_{A}\to\gamma(K,H)$, where $K$ is a separable Hilbert space, satisfy assumptions A.1 and A.2 with $\gamma$ independent of $M\in(0,1)$ satisfying

[TABLE]

*Then, for every $(u_{0},u_{1})\in H_{A}\times H$ satisfying condition (5.21), there exists a unique local maximal mild solution $u=\{u(t):t\in[0,\tau)\}$, to the problem (5.3), in the sense of Definition 5.12.

Moreover, there exists an announcing sequence $\{\tau_{k}\}_{k\geq 1}$ of the stopping time $\tau$, such that if $\mathbb{P}(\tau<\infty)>0$, then $\mathbb{P}$-almost surely on the set $\{\tau<\infty\}$, the following explosion condition is satisfied*

[TABLE]

$\phi$* is function defined by*

[TABLE]

Proof of Theorem 6.1.

The proof is divided into 3 parts. We prove the existence part in the first and the uniqueness result in the second part. We introduce the concept of maximality with its proof in our setting in the last part. For the purpose of the remaining proof we will consider a numerical sequence $M=(M_{n})_{n=2}^{\infty}=(1-\frac{1}{n})_{n=2}^{\infty}$.

6.2. A proof of the existence of a local solution

The proof of the existence of a local mild solution is carried out in three steps. The first two steps are devoted to prove the existence and the uniqueness of a solution of a truncated equation. In the third step we prove the existence of a local mild solution, in the sense of Definition 5.12, to problem (5.3).

Step I: Here we introduce the truncated evolution equation (6.2), related to problem (5.3). Then we prove some important estimates which will allow us in Step II to show the local well-posedness of equation (6.2).

Let us choose and fix the initial data $(u_{0},u_{1})\in H_{A}\times H$ satisfying condition (5.21). Since then $\|u_{0}\|_{H_{A}}$ is less than $1$, we can find $n\in\mathbb{N}\setminus\{1\}$ such that

[TABLE]

In the remainder of Step I we fix $n$ and the corresponding number $M_{n}$. We introduce the following auxiliary function. Let $\theta:\mathbb{R}_{+}\to[0,1]$ be a smooth function with compact support such that

[TABLE]

and for $n\geq 1$ set \theta_{n}(\cdot)=\theta\bigl{(}\frac{\cdot}{n}\bigr{)}.

The following lemma states the basic properties of $\theta_{n}$.

Lemma 6.2**.**

The functions $\theta_{n}$ is Lipschitz and bounded and, for all $x,y\in[0,\infty)$,

[TABLE]

Moreover, if $h:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a non decreasing function, then for all $x\in[0,\infty)$,

[TABLE]

Remark 6.3**.**

Let us point out that there is a typo in the lower bound of $\inf_{x\in\mathbb{R}_{+}}\theta$ in [11, (4.10)] and the value of this lower bound should be strictly smaller than $-1$. Indeed, we can easily show that there does not exist a smooth function satisfying all the specified conditions in [11, (4.10)]. Consequently, the Lipschitz constant in [11, Lemma  4.3] should be strictly greater than $\frac{1}{n}$.

Recall that, from Remark 5.2, for fixed $M_{n}$ as above and given maps $F$ and $G$ which satisfy (5.5) and (5.7), respectively, on an open ball $B(0,M_{n})$ of radius $M_{n}$ around [math] in space $H_{A}$, by Theorem E.3, there exist maps $\tilde{F}_{n}$ and $\tilde{G}_{n}$ (we do write the explicit dependency on $n$ for accuracy), defined on $H_{A}\cap E$, taking values in spaces $H$ and $\gamma(K,H)$, respectively, such that the inequalities (5.8) and (5.9) hold true for every $u,v\in H_{A}\cap E$.

Next, for $T>0$, we define a map

[TABLE]

by requesting that $\Psi_{T}^{n}(u)$ satisfies the following equation, for all $t\in[0,T]$,

[TABLE]

Remark 6.4**.**

It is important to note that the cut-off function $\theta_{n}$ plays essential role in the fixed point argument we display here because of the quasi-Lipschitz properties of $\tilde{F}$ and $\tilde{G}$, see (5.8) and (5.9), respectively.

Let us recall that the $Y_{t}$ norm has been defined in (4.7). We will show that there exists $T_{n}>0$ such that $\Psi_{T}^{n}$ is a strict contraction. We divide our argument in a couple of lemmata.

Lemma 6.5**.**

If $T>0$, then the map

[TABLE]

is well-defined and

[TABLE]

Proof of Lemma 6.5.

Let us choose and fix $T>0$. It is known that, see [2], $w:=\mathscr{I}_{1}^{n}(u_{0},u_{1})$ is the unique solution of the following homogeneous wave equation, with the Dirichlet or the Neumann boundary condition,

[TABLE]

Moreover, see Remark 3.3, $w$ belongs to \mathrm{C}\bigl{(}[0,T];H_{A}\bigr{)}=X_{T} with

[TABLE]

and, by Theorem 3.2, $w$ belongs to $Y_{T}$ and satisfy

[TABLE]

So, for every $\omega\in\Omega$, $\mathscr{I}_{1}^{n}(u_{0},u_{1})\in Y_{T}\cap X_{T}$ and (5.2) is satisfied. Furthermore, since the process $\mathscr{I}_{1}^{n}(u_{0},u_{1})$ is $\mathbb{F}$-adapted and continuous, it is progressively measurable and, hence Lemma 6.5 follows. We only need to observe that in view of (5.2) we have

[TABLE]

The proof of Lemma 6.5 is complete. ∎

Since by Assumptions A.1 and A.2 $F(0)=0$ and $G(0)=0$, we get $\tilde{F}_{n}(0)=0$ and $\tilde{G}_{n}(0)=0$. Moroever we deduce the following result.

Corollary 6.6**.**

If $\mathscr{I}_{1}^{n}$ is defined in Lemma 6.5 then

[TABLE]

Also, if $R>1$ and the initial data satisfies

[TABLE]

then,

[TABLE]

Proof.

Inequality (6.8) is a consequence of inequality (6.6). ∎

Lemma 6.7**.**

If $T>0$, then the map

[TABLE]

is well-defined and satisfies

[TABLE]

Proof of Lemma 6.7.

Let us choose and fix $T>0$. Take an arbitrary $u\in\mathbb{M}^{p}(Z_{T})$ and $\tilde{u}=\mathscr{I}_{2}^{n}(u)$. Then, by definition (6.7) of the map $\mathscr{I}_{2}^{n}$, we have

[TABLE]

Note that, in the last step we used the following bound, which is a consequence of [60, Lemma 2.2] applied to $C_{0}$-group associated to the wave operator $\partial_{tt}-\Delta$,

[TABLE]

where $K_{T}:=\tilde{M}e^{mT}$ for some constants $m\geq 0$ and $\tilde{M}\geq 1$. Let $T^{\ast}$ be stopping time defined by

[TABLE]

If the set in the definition of $T^{\ast}$ is empty, then we set $T^{\ast}=T$. Returning to (6.11), by applying (6.13) we get

[TABLE]

Moreover, since in view of (6.13), $\|u\|_{X_{T^{\ast}}},\|u\|_{Y_{T^{\ast}}}\leq 2n$ and since $\tilde{F}_{n}(0)=0$, by Lemma 5.3 we infer the following inequality

[TABLE]

where $C_{\tilde{F}_{n}}^{\prime}:=K(\gamma)C_{\tilde{F}_{n}}$. Combining (6.11), (6.14) and (6.2) we have

[TABLE]

By definition (6.7) of the map $\mathscr{I}_{2}^{n}$, invoking the inhomogeneous Strichartz estimates from Theorem 3.2 followed by (6.2) we get

[TABLE]

Which consequently, after taking the expectation, gives,

[TABLE]

Hence, since by estimates (6.16) and (6.17) we get the Lemma 6.7. ∎

The next result establishes the Lipschitz property of $\mathscr{I}_{2}^{n}$ as a map acting on $\mathbb{M}^{p}(Z_{T})$.

Lemma 6.8**.**

if $T>0$, then there exists a constant $L_{2}^{n}(T)>0$ such that the following assertions are true:

• •

for every $n\in\mathbb{N}$, $L_{2}^{n}(\cdot)$ is non decreasing;

• •

for every $n\in\mathbb{N}$, $\lim\limits_{T\to 0}L_{2}^{n}(T)=0$;

• •

for every $u_{1},u_{2}\in\mathbb{M}^{p}(Z_{T})$, $\mathscr{I}_{2}^{n}$ satisfies

[TABLE]

Proof of Lemma 6.8.

Let $u_{1},u_{2}\in\mathbb{M}^{p}(Z_{T})$. Since $\mathscr{I}_{2}^{n}$ is well-defined, we denote $\tilde{u}_{1}:=\mathscr{I}_{2}^{n}(u_{1}),\tilde{u}_{2}:=\mathscr{I}_{2}^{n}(u_{2})\in\mathbb{M}^{p}(Z_{T})$. As in the proof of Lemma 6.7, we define the following stopping times

[TABLE]

Let us observe that for $t\in[0,T]$, $\mathbb{P}$-a.s.,

[TABLE]

Moreover, similarly to inequality (6.2) we get

[TABLE]

Therefore, by invoking the inhomogeneous Strichartz estimates from Theorem 3.2, we get the following inequality

[TABLE]

where $\bar{C}_{T}^{p}:=2^{p-1}C_{T}^{p}$. Now, since $\theta_{n}(\|u_{1}\|_{Z_{s}})=0$ for $s\geq T_{1}$ and $\|u_{2}\|_{Z_{t}}\leq 2n$ for $t\in[0,T_{2}]$, $\mathbb{P}$-a.s., by using the Lemmata 5.3 and 6.2, we estimate the first of the two integrals in the right hand side above as

[TABLE]

Swapping between $u_{2}$ and $u_{1}$ we can analogously show the following estimate the second of the two integrals in the RHS of (6.2), i.e.

[TABLE]

Thus, by combining the computation from (6.2)-(6.2) we obtain

[TABLE]

Next, using the inequality (6.12), followed by repeating calculations as in (6.2) and (6.2), we obtain

[TABLE]

Hence,, in combination with the estimate (6.2) we get

[TABLE]

Since $\gamma<p$, by definition of $L_{2}^{n}(T)$, it is clear that, for each $n\in\mathbb{N}$, $\lim\limits_{T\to 0}L_{2}^{n}(T)=0$. Thus, the proof of Lemma 6.8 is complete. ∎

We continue with the proof of Theorem 6.1. We set

[TABLE]

Then we can write

[TABLE]

In the next result, we show that $\mathscr{I}_{3}^{n}$ maps $\mathbb{M}^{p}(Z_{T})$ into itself.

Lemma 6.9**.**

For any $T>0$, the map

[TABLE]

where $J\xi^{n}$ is as (6.2), is well-defined and satisfies

[TABLE]

Proof of Lemma 6.9.

Take any $u\in\mathbb{M}^{p}(Z_{T})$ and set $\tilde{u}:=J\xi_{u}^{n}$. Observe that from (4.24), we have

[TABLE]

Let us define $T^{\ast}:=\inf\{t\in[0,T]:\|u\|_{Z_{t}}\geq 2n\}$ as a stopping time. Since $u\in\mathbb{M}^{p}(Z_{T})$, the maps $t\mapsto\|u\|_{Z_{t}}$ is non-decreasing and continuous. Consequently, we infer that $\theta_{n}(\|u\|_{Z_{t}})=0$ for all $t\in(T^{\ast},T]$, and

[TABLE]

Thus, invoking Lemma 5.5, followed by the Hölder inequality give

[TABLE]

Here $C_{\tilde{G}_{n}}^{\prime}:=K(\gamma)C_{\tilde{G}_{n}}$. Consequently, by using (6.2) in (6.23) we obtain

[TABLE]

Next, to estimate $\mathbb{E}\left[\|J\xi_{u}^{n}\|_{X_{T}}^{p}\right]$, using the stochastic Strichartz estimates from Theorem 4.7, followed by (6.2), we get

[TABLE]

Combining (6.25) and (6.26) completes the proof of Lemma 6.9. ∎

The next result establishes the Lipschitz property of the map $\mathscr{I}_{3}^{n}$ defined in (6.22).

Lemma 6.10**.**

Assume that a Hilbert space $K$ satisfies Assumption (5.4). If $T>0$, then there exists a constant $L_{3}^{n}(T)>0$ such that the following assertions are true:

• •

$L_{3}^{n}(\cdot)$* is non decreasing;*

• •

for every $n\in\mathbb{N}$, $\lim\limits_{T\to 0}L_{3}^{n}(T)=0$;

• •

for $u_{1},u_{2}\in\mathbb{M}^{p}(Z_{T})$, $\mathscr{I}_{3}^{n}$ satisfy,

[TABLE]

Proof of Lemma 6.10.

Let us choose and fix $T>0$. We set

[TABLE]

Since $\mathscr{I}_{3}^{n}$ is well-defined, we denote $J\xi_{1}^{n}:=J\xi_{u_{1}}^{n},J\xi_{2}^{n}:=J\xi_{u_{2}}^{n}\in\mathbb{M}^{p}(Z_{T})$. Then, note that since for $t\in[0,T]$, $\mathbb{P}$-a.s.,

[TABLE]

Thus, applying (4.24) from Theorem 4.7 gives

[TABLE]

Now, we define the following stopping times

[TABLE]

By applying the stochastic Strichartz estimate (4.21) from Theorem 4.7, we get

[TABLE]

Next, observe that computation similar to (6.2) gives

[TABLE]

As in the proof of Lemma 6.8, we will only estimate the first of the two integrals in the right hand side above. Since $\theta_{n}(\|u_{1}\|_{Z_{t}})=0$ for $t\geq T_{1}$ and $\|u_{2}\|_{Z_{t}}\leq 2n$ for $t\in[0,T_{2}]$, $\mathbb{P}$-a.s., by following the computation of (6.2) and using the Lemmata 5.5 and 6.2, we estimate the first integral in the right hand side of (6.2) as

[TABLE]

Swapping between $u_{2}$ and $u_{1}$ we can analogously show the following estimate the second of the two integrals in the RHS of (6.2), i.e.

[TABLE]

Thus, substituting (6.2)-(6.2) into (6.29) and and (6.29) yield, respectively,

[TABLE]

Hence, we get

[TABLE]

Since $2\gamma<p$, by definition of $L_{3}^{n}(T)$, it is clear that $\lim\limits_{T\to 0}L_{3}^{n}(T)=0$ for every $n$. Thus, we have proved (6.27) and hence the proof of Lemma 6.10 is complete. ∎

Step II: Let us recall that $n\in\mathbb{N}$ and the positive numbers $M_{n}$ and $M_{n}^{\prime}$ are fixed such that (6.3) holds true. In this step, we prove the following auxiliary result.

Proposition 6.11**.**

There exists $T_{n}>0$ such that the map $\Psi_{T_{n}}^{n}$ defined by (6.4)-(6.2) has a unique fixed point in the space $\mathbb{M}^{p}(Z_{T_{n}})$.

Proof of Proposition 6.11.

From Lemmata 6.5 - 6.10, we infer that, for any $T>0$, the map $\Psi_{T}^{n}$ is well-defined on $\mathbb{M}^{p}(Z_{T})$ and for every $u_{1},u_{2}\in\mathbb{M}^{p}(Z_{T})$, we have

[TABLE]

where $L_{n}(\cdot)$ is non-decreasing and $\lim\limits_{T\to 0}L_{n}(T)=0$. Hence, we can find $T_{n}>0$ such that $\Psi_{T_{n}}^{n}$ is a $\frac{1}{2}$-contraction. Thus, by the Banach Fixed Point Theorem there exists a unique fixed point $u_{n}\in\mathbb{M}^{p}(Z_{T_{n}})$ of the map $\Psi_{T_{n}}^{n}$ and the proof of Proposition 6.11 is complete. ∎

Remark 6.12**.**

Since $L_{n}(T)\leq\frac{1}{2}$ we infer that

[TABLE]

Therefore, if the initial data satisfies assumption (6.7), then by inequality (6.8) in Corollary 6.6, the unique fixed point $u_{n}$ of $\Psi_{T}^{n}$ satisfies the following estimate

[TABLE]

where the constant $C(T)$ is increasing w.r.t. $T$ and is given by.

[TABLE]

Step III: Here we prove the existence of a local mild solution to problem (5.3) in the sense of Definition 5.12. Let $T_{n}>0$ be as in Proposition 6.11. Let $u_{n}\in\mathbb{M}^{p}(Z_{T_{n}})$ be unique fixed point of map $\Psi_{T_{n}}^{n}$. Let $\tau_{n}$ be a stopping time defined by the following formula

[TABLE]

where $\phi$ is function defined earlier in (6.2). In view of the definition (4.6) of the $\|\|_{X_{t}}$-norm we deduce the following equivalent definition of the stopping time $\tau_{n}$,

[TABLE]

At this juncture it is important to mention that, since $\|u_{n}(0)\|_{H_{A}}<M_{n}$, $\|u_{n}\|_{Z_{0}}=\|u_{n}(0)\|_{H_{A}}<1<n$ and the maps $t\mapsto\|u_{n}\|_{Z_{t}}$ and $t\mapsto\|u_{n}\|_{X_{t}}$ are continuous, the stopping time $\tau_{n}$ is strictly positive $\mathbb{P}$-almost surely.

Since our filtered probability space $(\Omega,\mathscr{F},\mathbb{P},\mathbb{F})$ satisfies the usual hypothesis, the notions of accessible and predictable stopping times are equivalent, see [47, Theorem I.6.6] and therefore by [47, Proposition I.4.14], the $\tau_{n}$ is an accessible stopping time. Thus, we can find an announcing sequence $\{\tau_{n_{k}}\}_{k\in\mathbb{N}}$ for $\tau_{n}$.

Now we claim that the local process $\tilde{u}_{n}$ defined by

[TABLE]

is a local solution. Since the accessibility of $\tau_{n}$ follows from above , we need to prove properties (2) and (3) of Definition 5.12 for $\{u(t);t\in[0,\tau_{n})\}$. The property $\|u(t)\|_{H_{A}}<1$ for $t\in[0,\tau_{n}),\mathbb{P}$-a.s. directly follows be definition of $\tau_{n}$ and the map $\phi$. Next, observe that since $u\in\mathbb{M}^{p}(Z_{T_{n}})$, $u$ belongs to $\mathbb{M}^{p}(Z_{t\wedge\tau_{n_{k}}})$, for all $t\geq 0$ and every $k$.

Since $u_{n}$ is the fixed point of map $\Psi_{T}^{n}$, it satisfies, for every $t\in[0,T]$, the following equation

[TABLE]

Observe that, since from the definition of $\mathbb{M}^{p}(Z_{T})$, the processes on both sides of equality (6.2) are $\mathscr{D}(A^{\frac{1}{2}})$-valued continuous, and by [63, Theorem I.2], any two modified stochastic processes are indistinguishable, we conclude that the equality even holds when the fixed deterministic time is replaced by the random one, in particular, (6.2) holds for $t\wedge\tau_{n_{k}}$. Moreover, since $\tau_{k}\leq\tau$, we infer that $\phi(\|u\|_{X_{t\wedge\tau_{n_{k}}}})\leq n$ for all $k\in\mathbb{N}$. Therefore, $\|u\|_{X_{t\wedge\tau_{n_{k}}}}\leq 1-\frac{1}{n}$ for all $k\in\mathbb{N}$ and

[TABLE]

Consequently, since by the definition of $\theta_{n}$ and $\tau_{n_{k}}\leq\tau$ we have $\theta_{n}(\|u_{n}\|_{Z_{t\wedge\tau_{n_{k}}}})=1$ for all $n,k\in\mathbb{N}$, we deduce that for every $t\geq 0$,

[TABLE]

Moreover, by invoking Lemma A.1, which is a generalization of [10, Lemma A.1], we deduce that for every $t\geq 0$,

[TABLE]

where $I_{\tau_{n_{k}}}(G(\tilde{u}_{n}))(t)$ is defined in (5.23). This proves that the equality (3) is satisfied by the process $\tilde{u}_{n}$ and hence concludes the proof of the existence part of Theorem 6.1.

6.3. A proof of the uniqueness of a local solution

The proof below is based on the proof of [9, Theorem 5.14].

Theorem 6.13**.**

Assume that $(u_{0},u_{1})\in H_{A}\times H$ and that condition (5.21) is satisfied. Assume that $(u,\tau)$ and $(v,\sigma)$ are two local solutions of problem (5.3), with the same initial data $(u_{0},u_{1})$. Then,

[TABLE]

Proof of Theorem 6.13.

Let us choose and fix two local solutions $(u,\tau)$ and $(v,\sigma)$ of problem (5.3), respectively, with the same fixed initial data $(u_{0},u_{1})\in H_{A}\times H$ satisfying condition (5.21). Without loss of generality, we can assume that $\tau\vee\sigma\leq T$ for some $T>0$ which we fix for the remaining of the proof.

Let $\{\tau_{n}\}_{n\in\mathbb{N}}$ and $\{\sigma_{n}\}_{n\in\mathbb{N}}$ be the announcing sequences of $\tau$ and $\sigma$, respectively. Since the considered probability space satisfies the usual hypothesis, by [47, Propositions 4.3 and 4.11, and Theorem 6.6] the stopping time $\varrho:=\tau\wedge\sigma$ is accessible and it is easy to show that $\{\varrho_{n}\}_{n\in\mathbb{N}}:=\{\tau_{n}\wedge\sigma_{n}\}_{n\in\mathbb{N}}$ is an announcing sequence of $\varrho$.

Let us fix $n\in\mathbb{N}$. Since $(v,\sigma)$ is a local solution to the problem (5.3) we have that, for all $t\in[0,T]$,

[TABLE]

where

[TABLE]

Since the above holds true for for all $t\in[0,T]$, by replacing $t$ by $t\wedge\varrho_{n}$ we get, $\mathbb{P}$-a.s.,

[TABLE]

Consequently, since $\varrho_{n}\leq\sigma_{n}$, Lemma (A.3) yields, for every $t\in[0,T]$, $\mathbb{P}$-almost surely

[TABLE]

This proves that $(v,\varrho)$ is a local solution to problem (5.3). In a similar way, we can also prove that $(u,\varrho)$ is a local solution to problem (5.3).

Let us choose and fix an arbitrary $M\in(0,1)$ such that $\|u_{0}\|_{H_{A}}<M$ and fix $k\in\mathbb{N}$. We define the following six stopping times

[TABLE]

Arguing as in the proof of (6.3) we can show that for all $t\in[0,T]$, $\mathbb{P}$-a.s.

[TABLE]

Similarly, we can prove that the above identity holds with $v$ replaced by $u$. Hence, by setting $w=u-v$ we obtain, for all $t\in[0,T]$, $\mathbb{P}$-a.s.,

[TABLE]

where

[TABLE]

In order to move forward, we set

[TABLE]

It is obvious that $w_{0}=w_{1}=0$. Observe that

[TABLE]

Invoking the inequality (3.6) and the stochastic Strichartz estimates (4.27), with $p=2$, and observing that the inequalities (3.6) and (4.27) hold for derivative also, followed by the Assumptions A.1 and A.2 we infer that,

[TABLE]

Now we can apply Lemma D.1 with the following choice of processes.

[TABLE]

Note that by the definition of the stopping time $\varrho_{n,k,M}$ the following inequality holds and therefore Assumption (D.1) from Lemma D.1 is satisfied.

[TABLE]

Finally, inequality (6.39) holds for all possible pairs $(\tau_{a},\tau_{b})$ of accessible stopping times such that $0\leq\tau_{a}\leq\tau_{b}\leq\varrho_{n,k,M}$. Thus, since $w(0)=0$, by Lemma D.1 we infer that

[TABLE]

This implies that there exists $\Omega_{n,k,M}\in\mathscr{F}$ such that $\mathbb{P}(\Omega_{n,k,M})=1$ and

[TABLE]

Put $\Omega_{k,M}=\cup_{n}\Omega_{n,k,M}$. Note that $\mathbb{P}(\Omega_{k,M})=1$. Since $\tau_{n}\wedge\sigma_{n}\nearrow\tau\wedge\sigma$ we infer that

[TABLE]

Put $\Omega_{0}:=\cup_{M\in(0,1)\cap\mathbb{Q}}\cup_{k}\Omega_{k,M}$. Note that $\mathbb{P}(\Omega_{0})=1$. Since $\lim_{M\nearrow 1}\lim_{k\to\infty}\varrho_{k,M}=\tau\wedge\sigma$, $\mathbb{P}$-almost surely, we infer that there exists $\Omega_{1}\in\mathscr{F}$ such that $\mathbb{P}(\Omega_{1})=1$ and $\varrho_{k,M}\nearrow\tau(\omega)\wedge\sigma(\omega)$ for each $\omega\in\Omega_{1}$. Hence we infer that

[TABLE]

Because $\mathbb{P}(\Omega_{0}\cap\Omega_{1})=1$, this completes the proof of uniqueness.

∎

Lemma 6.14**.**

For every $n\in\mathbb{N}$, we have $\tau_{n}\leq\tau_{n+1},\mathbb{P}$-a.s. and for every $t\in[0,\tau_{n})$, we have $u^{n}(t)=u^{n+1}(t),\mathbb{P}$-a.s..

Proof of Lemma 6.14.

Let us fix any $n\in\mathbb{N}$. We will prove it by contradiction. So suppose that there exists a $\Omega_{0}\subset\Omega$ such that $\mathbb{P}(\Omega_{0})>0$ and $\tau_{n}(\omega)>\tau_{n+1}(\omega)$ for every $\omega\in\Omega_{0}$. Wlog we can assume that $\mathbb{P}(\Omega_{0})=1$ and $\tau_{n}(\omega)>\tau_{n+1}(\omega),\mathbb{P}$a.s.. This implies that $\tau_{n}\wedge\tau_{n+1}=\tau_{n+1},\mathbb{P}$-a.s and consequently, because of our uniqueness result, which is Theorem 6.13, we get

[TABLE]

Recall that $[0,\tau_{n+1})\times\Omega_{0}=\{(t,\omega)\in[0,\infty)\times\Omega_{0}:0\leq t<\tau_{n+1}(\omega)\}.$ Then, observe that by Definition 2.15 of equivalence of two local stochastic processes and the continuity of solutions $u^{n+1},u^{n}$ we obtain

[TABLE]

Here we get the last inequality because $(u^{n},\tau_{n})$ is a local solution and $\tau_{n}<\tau_{n+1},\mathbb{P}$-a.s.. But the above inequality is absurd and hence we get a contradiction about the existence of set $\Omega_{0}$ of positive probability. So we have proved that $\tau_{n}\leq\tau_{n+1},\mathbb{P}$-a.s.. This also implies that, due to Definition 2.15, the equality $u^{n}=u^{n+1}$ on $[0,\tau_{n}),\mathbb{P}$-a.s., since $\Omega_{t}(\tau_{n})\cap\Omega_{t}(\tau_{n+1})=\Omega_{t}(\tau_{n})$. Hence the Lemma 6.14. ∎

6.4. Proof of the existence of local maximal solution

In this subsection we complete the proof of Theorem 6.1 by proving the existence of a unique maximal local solution. The proof of the following lemma is a slight modification of [9, Lemma 5.3], see also [33, Lemmata 6A and 6B] and [30, Chapter 5, Section 18].

Lemma 6.15**.**

**(The Amalgamation Lemma) **

• (1)

Let $\Delta$ be a family of accessible stopping times taking values in $[0,\infty]$ such that the supremum of every finite subset of $\Delta$ belongs to $\Delta$. Then supremum $\tau$ of $\Delta$ is a $[0,\infty]$-valued accessible stopping time and there exists an increasing sequence $\{\alpha_{n}\}_{n=1}^{\infty}\subset\Delta$ of accessible stopping times such that $\tau(\omega)=\lim_{n\to\infty}\alpha_{n}(\omega)$, for all $\omega\in\Omega$.

• (2)

Assume also that for each $\alpha\in\Delta$, $I_{\alpha}:[0,\alpha)\times\Omega\to E$ is a local progressively measurable such that for all $\alpha,\beta\in\Delta$ and every $t>0$,

[TABLE]

Then, there exists a progressively measurable process $I:[0,\tau)\times\Omega\to E$, such that for every $t>0$,

[TABLE]

• (3)

Moreover, if ${\tilde{I}}:[0,\tau)\times\Omega\to E$ is any local process satisfying (6.40) then the process $\tilde{I}$ is a version of the process $I$, that is, for all $t\in[0,\infty)$

[TABLE]

Proposition 6.16**.**

Suppose that the assumptions of Theorem 6.1 hold. Assume also the following two conditions:

• (i)

there exist at least one local solution $(u^{0},\tau^{0})$ to problem (5.3);

• (ii)

if $(u^{1},\tau^{2})$ and $(u^{2},\tau^{2})$ are local solutions to problem (5.3), then for every $t>0$,

[TABLE]

Then, the following assertions hold.

• (1)

If $(u^{1},\tau^{1})$ and $(u^{2},\tau^{2})$ are two local solutions to problem (5.3), then the process $(\tilde{u},\tilde{\tau})$ defined by

[TABLE]

is a local solution to problem (5.3). Here $\Omega_{1}:=\{\tau^{1}\geq\tau^{2}\}$ and $\Omega_{2}:=\{\tau^{2}>\tau^{1}\}$.

• (2)

problem (5.3) has a unique maximal local solution $(\hat{u},\hat{\tau})$ satisfying $(u^{0},\tau^{0})\preceq(\hat{u},\hat{\tau})$.

Proof of Proposition 6.16.

Here we closely follow the proof of [9, Proposition 5.7]. To prove the first assertion, let us suppose that $(u^{1},\tau^{1})$ and $(u^{2},\tau^{2})$ are two local solutions to problem (5.3). Let $\{\tau^{i}_{n}\}_{n\in\mathbb{N}}$, $i=1,2$ be the announcing sequence of $\tau^{i}$ and set $\tilde{\tau}:=\tau^{1}\vee\tau^{2}$. Then, observe that $\tilde{\tau}$ is an accessible stopping time with announcing sequence $\{\tilde{\tau}_{n}:=\tau^{1}_{n}\vee\tau^{2}_{n}\}_{n\in\mathbb{N}}$. By the assumption (ii), we deduce that

[TABLE]

that is, for all $t\geq 0$,

[TABLE]

But since for $\omega\in\Omega_{t}(\tau^{1}\wedge\tau^{2})$ we always have $t<\tau^{1}\wedge\tau^{2}$ and we can write above as

[TABLE]

To show that the process $(\tilde{u},\tilde{\tau})$ is a local solution to problem (5.3), without loss of generality, we assume that $\tau^{1}_{n}(\omega)\leq\tau^{2}_{n}(\omega)$ for all $\omega\in\Omega$ and $n\in\mathbb{N}$. This implies that $\tilde{\tau}_{n}=\tau_{n}^{2}$ on $\Omega$. Let us fix $n\in\mathbb{N}$ and $t\geq 0$. Note that the proof of the progressive measurability of $\{\tilde{u}(t),\;t\in[0,\tilde{\tau})\}$ and the continuity of its paths is very similar to the proof in [6, Corollary 2.28]. Next, we observe that on $\Omega_{1}$ we have $\tilde{\tau}_{n}<\tau^{1}\wedge\tau^{2}=\tau^{2}$. Hence, on $\Omega_{1}$ from (6.43) and (6.41) we deduce that,

[TABLE]

where

[TABLE]

The last equality follows from the fact that $(u^{2},\tau^{2})$ is a local solution. Since $\tau_{n}^{2}=\tilde{\tau}_{n}$, by using (6.41), (6.42) and [6, Proposition 2.10] we deduce that

[TABLE]

where

[TABLE]

Hence, $(\tilde{u},\tilde{\tau})$ satisfies equation (3) on $\Omega_{1}$. In a similar way, we can also show that $(\tilde{u},\tilde{\tau})$ satisfies (3) on $\Omega_{2}$. Since $\Omega_{1}\cup\Omega_{2}$ is of measure $1$, we get that $(\hat{u},\hat{\tau})\in\mathcal{LS}$ and hence the assertion (1).

To prove the second assertion, let us choose and fix a local solution $(u^{0},\tau^{0})$ to problem (5.3). Let us consider the subset of $\mathcal{LS}$ which consists of all local solution $(u,\tau)$ to the problem (5.3) such that $(u^{0},\tau^{0})\preceq(u,\tau)$. Note that by assumption (i) of Proposition 6.16 this set is non-empty. Due to the assumptions (i), (ii) and assertion (1) of Proposition 6.16, we apply the Amalgamation Lemma 6.15 and infer that there exists an accessible stopping time

[TABLE]

and a progressively measurable process $\hat{u}:[0,\hat{\tau})\times\Omega\to E$, having continuous trajectories such that for all $(u,\tau)\in\mathcal{LS}$ and for $t>0$,

[TABLE]

Moreover, there exists an increasing sequence $\{\hat{\tau}_{n}\}_{n\in\mathbb{N}}$ of accessible stopping times such that $\hat{\tau}(\omega)=\lim_{n\to\infty}\hat{\tau}_{n}(\omega)$, for all $\omega\in\Omega$.

To prove the existence of a maximal local solution, it is sufficient to show that $(\hat{u},\hat{\tau})\in\mathcal{LS}$. For this objective, let us define an auxiliary process $\hat{\eta}=\{\hat{\eta}(t):t\in[0,\hat{\tau})\}$, such that for each $n\in\mathbb{N}$ and $t\geq 0$, the following equality holds $\mathbb{P}$-a.s.

[TABLE]

where $I_{\tau_{n}}(G)$ is a continuous $E$-valued process defined by

[TABLE]

Assume that $(u,\tau)\in\mathcal{LS}$, which is possible due to the assumption (i) in the statement of the current proposition. Define a process $\eta=\{\eta(t):t\in[0,\tau)\}$ by formulae (6.4)-(6.46) with $\hat{u}$ replaced by $u$ and the announcing sequence $\{\hat{\tau}_{n}\}_{n\in\mathbb{N}}$ of the accessible stopping time $\hat{\tau}$ replaced by announcing sequence $\{\tau_{n}\}_{n\in\mathbb{N}}$ of the accessible stopping time $\tau$. Since $(u,\tau)$ is a local solution, we infer that the process $\{\eta(t):t\in[0,\tau)\}$ is a version of the process $\{u(t):t\in[0,\tau)\}$. Because $\hat{u}$ satisfies (6.44), we have, for each $s\leq t$,

[TABLE]

and due to [6, Proposition 2.10], see also [35], we infer that

[TABLE]

Hence, the process $\hat{\eta}$ satisfy (6.40) and then by the part (3) of Lemma 6.15, we infer that the process $\{\hat{\eta}(t):t\in[0,\hat{\tau})\}$, is a version of the process $\{\hat{u}(t):t\in[0,\hat{\tau})\}$ and therefore we can replace $\hat{\eta}$ by $\hat{u}$ on the LHS of (6.4). Therefore, we deduce that $(\hat{u},\hat{\tau})\in\mathcal{LS}$. This completes the existence of a local maximal solution.

It remains to prove the uniqueness of the local maximal solution. For that let us consider that $(u^{1},\tau^{1})$ and $(u^{2},\tau^{2})$ are two local maximal solutions to problem (5.3). Then, by the proof of assertion (1), for $i\in\{1,2\}$, $(u^{i},\tau^{i})\preceq(\tilde{u},\tilde{\tau})$ and there exists $i_{0}\in\{1,2\}$ such that $(u^{i_{0}},\tau^{i_{0}})\not\sim(\tilde{u},\tilde{\tau})$. But this contradicts the maximality of $(u^{i_{0}},\tau^{i_{0}})$ and hence we finish the proof of Proposition 6.16. ∎

Since due to subsections 6.2 and 6.3, the assumptions of Theorem 6.16 are satisfied and hence the proof of the first part of Theorem 6.1 is complete. The last part of the Theorem is proved in the following subsection. ∎

6.5. Explosion for a local maximal solution

The proof of the part of the Theorem 6.1 is straightforward. Let $(\hat{u},\hat{\tau})$ be the unique local maximal solution to the problem (5.3) whose existence we proved in previous Steps. For $n\in\mathbb{N}$ let $(\tilde{u}_{n},\tau_{n})$ be the unique local solution to the problem (5.3) defined by in (6.36). Firstly we observe that by Lemma 6.14

[TABLE]

in the sense of Definition 5.14. Hence in view of equality (6.35) we infer that

[TABLE]

This proves the assertion (6.1) from Theorem 6.1. The proof is complete.

Appendix A Stopped processes

In this appendix we justify the choice of $I_{\tau}(G)$ process in the Definition 5.12. The proof of the next result to some extent is analogous to the proof of [10, Lemma A.1], where this is formulated in terms of a semigroup.

Lemma A.1**.**

Assume that a process $\xi$ belongs to $\mathcal{M}^{p}(\mathcal{L}^{2}[0,T],\gamma(K,H))$. Set

[TABLE]

and

[TABLE]

For any stopping time $\tau$ and for all $t\in[0,T]$, the following holds

[TABLE]

Remark A.2**.**

Let us note that, since $\tau$ is a stopping time, due to [47, Theorem 1.6 and Proposition 4.2] the stochastic process $\mathds{1}_{[0,\tau)}$ is progressively measurable. In particular, the integrand in (A.1) is progressively measurable.

Note that, it follows from Lemma A.1 that if $\xi=0$ on $[0,\tau)$, then $I(t\wedge\tau)=0$ for all $t\in[0,T]$, $\mathbb{P}$-a.s.. It is relevant to mention that the importance of such results goes back to [4], [7], and [22]. The next result is useful in the proof of Theorem 6.13. We ask the reader to see [9, Corollary A.2] for the proof.

Lemma A.3**.**

Let $\xi\in\mathcal{M}^{p}(\mathcal{L}^{2}[0,T],\gamma(K,H))$ and $\tau,\sigma$ be two stopping times such that $\sigma\leq\tau$, then

[TABLE]

Appendix B On pointwise evaluation

Let us now formulate a special case of [38, Proposition 1.2.25]. This result is a converse to [31, Proposition 3.19] and is closely related to [13, Proposition B.4].

Proposition B.1**.**

Assume that $(\Omega,\mathscr{F},\mathbb{F},\mathbb{P})$, where $\mathbb{F}=(\mathscr{F}_{t}:t\geq 0)$ is a filtered probability space. Assume also that $E$ is a separable Banach space, $T>0$ and $p\in[1,\infty)$. Assume finally that

[TABLE]

Then, there exists a $\mathscr{B}([0,T])\otimes\mathscr{F}_{T}$/$\mathscr{B}(E)$-measurable function

[TABLE]

and there exists $\Omega^{\prime}\in\mathscr{F}_{T}$ such that $\mathbb{P}(\Omega^{\prime})=1$ and for every $\omega\in\Omega^{\prime}$, the following equality holdsIIIIII This equality is again imprecise. Rigorously, one should replace it by $[f(\cdot,\omega)]=F(\omega)\mbox{ in }L^{p}(0,T;E)$, where $[\cdot]$ is the equivalence class w.r.t. the Lebesgue measure $\mathrm{Leb}$. But, see Remark 4.3, it is standard to use this imprecise formulation.

[TABLE]

Moreover, if $g:[0,T]\times\Omega\to E$ is another $\mathscr{B}([0,T])\otimes\mathscr{F}_{T}$/$\mathscr{B}(E)$-measurable function such that the above assertion holds, then $f=g$, $\mathrm{Leb}\otimes\mathbb{P}$-almost everywhere.

Appendix C About the definition of a solution

Here we state an equivalence, without proof, between two natural definitions of a mild solution for SPDE (1.2). We begin by recalling the framework from Section 5. In particular, we set

[TABLE]

where $(p,q,r)$ is any suitable triple which satisfy (3.5). Let us recall that the linear (unbounded) operator $\mathcal{A}$ in the space $\mathfrak{H}:=H_{A}\times H$ and the $C_{0}$-group on $\mathcal{H}$ generated by it have been defined in formulae (4.32) and (4.33), respectively. Let us also recall that the space $\mathbb{M}^{p}(Z_{T})$ has been defined in (5.2). We assume that the maps $F$ and $G$ satisfy A.1 and A.2, respectively.

Proposition C.1**.**

Suppose that $u_{0}\in H_{A}$, $u_{1}\in H$, and $T>0$.

• •

If an $H_{A}$-valued process $u=\{u(t):t\in[0,T]\}$ such that $u\in\mathbb{M}^{p}(Z_{T})$, is a mild to problem (5.3), i.e. for all $t\in[0,T]$, $\mathbb{P}$-a.s

[TABLE]

Then, $\mathbb{P}$-almost surely, the process $u$ is differentiable as $H$-valued process and the $H_{A}\times H$-valued $\mathbb{F}$-adapted process $\mathfrak{u}$ defined by

[TABLE]

where $v(t)=u^{\prime}(t)$, $t\in[0,T]$, is a continuous $H$-valued process and satisfies, for all $t\in[0,T]$, $\mathbb{F}$-almost surely, the following equation,

[TABLE]

where $\mathfrak{u}(0)=(u_{0},u_{1})\in H_{A}\times H$, and

[TABLE]

• •

Conversely, if an $\mathbb{F}$-adapted continuous $H_{A}\times H$-valued process

[TABLE]

such that $u\in\mathbb{M}^{p}(Z_{T})$, is a solution to (C.2) with notation (C.3), then the process $u$ is a solution to (• ‣ C.1).

Proposition C.2**.**

Suppose that $u_{0}\in H_{A}$, $u_{1}\in H$, and $\tau$ is an accessible stopping time with an announcing sequence $\{\tau_{k}\}_{k\in\mathbb{N}}$.

• •

If an $H_{A}$-valued local process $u=\{u(t):t\in[0,\tau)\}$ is a local mild to problem (5.3), then, $\mathbb{P}$-almost surely, the trajectories of $u$ are differentiable and the $H_{A}\times H$-valued $\mathbb{F}$-adapted local process $\mathfrak{u}$ defined by

[TABLE]

where $v(t)=u^{\prime}(t)$, $t\in[0,\tau)$, is continuous and, for all $t\geq 0$, satisfies $\mathbb{P}$-almost surely, the following equations, for every $k\in\mathbb{N}$,

[TABLE]

where $\mathfrak{u}(0)=(u_{0},u_{1})\in H_{A}\times H$, the process is defined by

[TABLE]

and $\mathfrak{F}$ and $\mathfrak{G}$ are defined in (C.3).

• •

Conversely, if an $\mathbb{F}$-adapted continuous $H_{A}\times H$-valued local process \mathfrak{u}(t)=\bigl{\{}(u(t),v(t)), t\in[0,\tau)\bigr{\}}, such that

[TABLE]

satisfies, for all $t\geq 0$, $\mathbb{P}$-almost surely, for every $k\in\mathbb{N}$, equation (C.4) with the notation (C.3) and (C.5), then the process u=\bigl{\{}u(t):t\in[0,\tau)\bigr{\}} is a solution to (• ‣ C.1).

Appendix D Stochastic Gronwall lemma

The following result is a slight simplification of [36, Lemma 5.3] which in turn is a generalization of [32, Lemma 3.9].

Lemma D.1**.**

Let us assume that $\tau$ is an accessible bounded stopping time. Let $X$ and $R$ be real valued non-negative local processes defined on $[0,\tau)\Omega$ such that for some $\kappa>0$

[TABLE]

and $\mathbb{E}\int_{0}^{\tau}R(s)X(s)\,ds<\infty$. Suppose also that there exists $C_{0}>0$ such that for all pairs $(\tau_{a},\tau_{b})$ of accessible stopping times such that $0\leq\tau_{a}<\tau_{b}\leq\tau$,

[TABLE]

Then, there exists a constant $C=C(C_{0},T,\kappa)$ such that

[TABLE]

Proof of Lemma D.1.

The proof is almost identical to the proof of [36, Lemma 5.3]. Since our assumptions are weaker, as we only consider pairs of accessible stopping times, we only need to observe that in view of the assumption (D.1) there exist a natural number $N$ and a sequence $(\tau_{i})_{i=0}^{N}$ of accessible stopping times defined as follows: $\tau_{0}=0$ and

[TABLE]

If $\tau_{1}=T$, then we put $\tau_{j}=T$ for all $j\geq 2$. If $\tau_{1}<T$, then we proceed by induction.

Note that, it easily follows from the definition of $\tau_{j}$, $N$ indeed exists and satisfies $N\leq 2\kappa C_{0}$. ∎

Appendix E Extensions of functions

The following result is a slight generalization of a result that has been proved in [5].

Lemma E.1**.**

Let $X$ and $E$ be normed vector spaces with norms denoted respectively by $\|\,.\,\|_{X}$ and $\|\,.\,\|_{E}$. For $R>0$ define $\psi_{R}:X\cap E\to X\cap E$ through the formula

[TABLE]

Then for all $x,x_{1},x_{2}\in X\cap E$

[TABLE]

In particular, the range of $\psi_{R}$ is contained in the set $B_{X}(0,R)\cap E$.

The next result was also formulated and proved in [5].

Corollary E.2**.**

Let $X$ and $Y$ be normed vector spaces with norms denoted by $\|\,.\,\|$. Suppose that a map $g:X\to Y$ is Lipschitz on the closed ball $\bar{B}(0,R)$, $R>0$, with Lipschitz constant $C$. Then, there exists a bounded map $\tilde{g}:X\to Y$ such that $\tilde{g}=g$ on $\bar{B}(0,R)$ and $\tilde{g}$ is Lipschitz on $X$, with Lipschitz constant $3C$.

We conclude this section with a new result which tells about the existence of a quasi-Lipschitz extension with the Lipschitz constant being $3$ times the Lipschitz constant of the original quasi-Lipschitz map. This atypical extension type theorem is a generalization of [5, Corollary 3] and to be best knowledge of the authours, this is a new result.

Theorem E.3**.**

Let $X$, $E$ and $Y$ be normed vector spaces with norms denoted respectively by $\|\,.\,\|_{X}$, $\|\,.\,\|_{E}$ and $\|\,.\,\|_{Y}$. Suppose that a map

[TABLE]

where $\bar{B}_{X}(0,R)$ is the closed ball in $X$ centered at the origin and of radius $R$, satisfies the following “Lipschitz” property. There exists $C>0$ and $p\geq 0$ such that

[TABLE]

Then, there exists a map

[TABLE]

such that $\tilde{g}=g$ on $\bar{B}_{X}(0,R)$ and

[TABLE]

Proof of Theorem E.3.

Set

[TABLE]

Since the range of function $\psi_{R}$ is contained in the set $B_{X}(0,R)\cap E$, the function $\tilde{g}:X\cap E\to Y$ is well defined and coincides with $g$ on the set $\bar{B}_{X}(0,R)\cap E$.

Let us choose and fix for $x_{1},x_{2}\in X\cap E$. Then the generalized Lipschitz property (E.5) of $g$, the Lipschitz, w.r.t. the $X$-norm, property (E.2) of the function $\psi_{R}$ and the linear growth, w.r.t. the $E$-norm, property (E.4) of the function $\psi_{R}$ imply that

[TABLE]

This implies the inequality (E.6) and hence completes the proof of Theorem E.3. ∎

Acknowledgements: The authors would like to thank Jan van Neerven for discussion related to measurability part in the proof of auxiliary result Lemma 4.5, and to Mark Veraar for having useful conversations on the stochastic Fubini theorem and to provide a suitable reference for the inequality (4.10). They want to thank Martin Ondreját for useful discussions about stopping time. They are thankful to Gaurav Dhariwal and Anupam Gumber for careful reading of manuscript. The authors are also grateful to Fabian Hornung and Tomasz Kosmala for providing comments which led to improvement of the presentation of this paper. Finally, they would like to thank an anonymous referee, whose critical comments have contributed to better and clearer proofs. The second author wishes to thank the York Graduate Research School, to award the Overseas scholarship (ORS), and the Department of Mathematics, University of York, to provide financial support and excellent research facilities during the period of this work. The second author also would like to acknowledge the German Science Foundation DFG to provide the financial support, through the Research Unit FOR 2402, during the work on revision of this manuscript.