Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator

TL;DR

This paper investigates the long-term behavior of solutions to logistic equations involving the fractional Laplacian, demonstrating convergence to obstacle problem solutions and analyzing eigenvalue properties under perturbations.

Contribution

It provides new asymptotic results for fractional Laplacian logistic equations and establishes the cut-off property for eigenvalues with perturbed potentials.

Findings

Solutions converge strongly in energy space

Uniform convergence to obstacle problem solutions

Eigenvalues exhibit cut-off property under perturbations

Abstract

We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove strong convergence of solutions in the energy space and uniform convergence to the solution of an obstacle problem. As a by-product, we also prove the cut-off property for eigenvalues of the Dirichlet fractional Laplace operator perturbed by exploding potentials.

Equations

$$\left\{\begin{array}[]{l}\frac{dv_{p}}{dt}-\Delta^{\alpha}v_{p}=av_{p}-bv_{p}^{p},\quad\mbox{in}\,\,\,D\times(0,\infty),\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \,v_{p}=0,\quad\mbox{in}\,\,\,({\mathbb{R}}^{d}\setminus D)\times(0,\infty),\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \,v_{p}(0,\cdot)=\varphi,\quad\mbox{in}\,\,\,D.\end{array}\right.$$

$$\underset{x\in K}{\inf }b(x)>0.$$

$${\mathbb{I}}_{D\setminus {\overline{D}}_0}=\infty 1_{D_0}+1_{D\setminus {\overline{D}}_0}.$$

$$\left\{\begin{array}[]{l}\max\big{\{}\frac{dv}{dt}-\Delta^{\alpha}v-av,v-\mathbb{I}_{D\setminus\overline{D}_{0}}\big{\}}=0,\quad\mbox{in}\,\,\,D\times(0,\infty),\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \,v=0,\quad\mbox{in}\,\,\,D^{c}\times(0,\infty),\vskip 6.0pt plus 2.0pt minus 2.0pt\\ v(0,\cdot)=\varphi,\quad\mbox{in}\,\,\,D,\end{array}\right.$$

$$\underset{\delta \le t\le T}{\sup }\Vert v_p(t)-v(t){\Vert }_{\infty }+{\int }_0^T\Vert v_p(t)-v(t){\Vert }_{H_0^{\alpha }(D)}dt\to 0\text{mbox}asp\to \infty .$$

$$\left\{\begin{array}[]{l}\max\big{\{}-\Delta^{\alpha}u-au,u-\mathbb{I}_{D\setminus\overline{D}_{0}}\big{\}}=0,\quad\mbox{in}\,\,\,D,\vskip 6.0pt plus 2.0pt minus 2.0pt\\ u=0,\quad\mbox{in}\,\,\,D^{c},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ u>0,\quad\mbox{on}\,\,\,D,\end{array}\right.$$

$$\Vert v(t)-u{\Vert }_{\infty }+\Vert v(t)-u{\Vert }_{H_0^{\alpha }(D)}\to 0\text{mbox}ast\to \infty .$$

$$\left\{\begin{array}[]{l}-\Delta^{\alpha}u_{p}=au_{p}-bu_{p}^{p},\quad\mbox{in}\,\,\,D,\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \,u=0,\quad\mbox{in}\,\,\,D^{c},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \,u_{p}>0\quad\mbox{on}\,\,\,D\end{array}\right.$$

$$\Vert v_p(t)-u_p{\Vert }_{\infty }+\Vert v_p(t)-u_p{\Vert }_{H_0^{\alpha }(D)}\to 0\text{mbox}ast\to \infty .$$

$$\Vert u_p-u{\Vert }_{\infty }+\Vert u_p-u{\Vert }_{H_0^{\alpha }(D)}\to 0\text{mbox}asp\to \infty ,$$

$${\lambda }_1^D(-{\Delta }^{\alpha }+q_k)\nearrow {\lambda }_1^V\text{mbox}ask\to \infty .$$

$$\forall K\subset D\setminus \overline{V},K\text{mbox}-compact\underset{x\in K}{\inf }{q}_k(x)\nearrow \infty .$$

$$-Au(x)={\int }_{D}J(x,y)(u(x)-u(y))dy$$

$${\Delta }^{\alpha }u(x)$$

$$=\frac{c_{\alpha ,d}}{2}{\int }_{{\mathbb{R}}^d}\frac{u(x+y)+u(x-y)-2u(x)}{|y{|}^{d+2\alpha }}dy$$

$$D(\mathcal{E})=\{u\in L^2({\mathbb{R}}^d;m):{\int }_{{\mathbb{R}}^d}|\xi {|}^{2\alpha }|\hat{u}(\xi ){|}^{2}d\xi <\infty \},\mathcal{E}(u,v)={\int }_{{\mathbb{R}}^d}|\xi {|}^{2\alpha }\hat{u}(\xi )\bar{\hat{v}}(\xi )d\xi .$$

$$H^{\alpha }({\mathbb{R}}^d)=\{u\in L^2({\mathbb{R}}^d;m):\Vert u{\Vert }_{H^{\alpha }({\mathbb{R}}^d)}:=[u{]}_{H^{\alpha }({\mathbb{R}}^d)}^{1/2}+\Vert u{\Vert }_{L^2({\mathbb{R}}^d;m)}<\infty \}$$

$$[u{]}_{H^{\alpha }({\mathbb{R}}^d)}:={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{d+2\alpha }}dxdy.$$

$${\cal E}(u,v)=c_{\alpha,d}\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}\big{(}u(x)-u(y)\big{)}\big{(}v(x)-v(y)\big{)}|x-y|^{-d-2\alpha}\,dx\,dy,\quad u,v\in D({\cal E}).$$

$$\mathcal{E}(u,v)=(-Au,v),u\in D(A),v\in D(\mathcal{E}).$$

$$Au(x)={\Delta }^{\alpha }u(x),\text{mbox}a.e.x\in {\mathbb{R}}^d.$$

$$D({\mathcal{E}}_D)=\{u\in D(\mathcal{E}):\tilde{u}=0\mathcal{E}\text{mbox}-q.e.\text{mbox}on{\mathbb{R}}^d\setminus D\},{\mathcal{E}}_D(u,v)=\mathcal{E}(u,v),u,v\in D({\mathcal{E}}_D).$$

$$D(A_D)\subset D({\mathcal{E}}_D),{\mathcal{E}}_D(u,v)=(-A_{D}u,v),u\in D(A_D),v\in D({\mathcal{E}}_D)$$

$$({\Delta }^{\alpha }{)}_{|D}:=A_D.$$

$$H_0^{\alpha }(D):={\overline{C_c^{\infty }(D)}}^{H^{\alpha }(D)}={\overline{C_c^{\infty }(D)}}^{H^{\alpha }({\mathbb{R}}^d)}.$$

$$D({\mathcal{E}}_D)={\overline{C_c^{\infty }(D)}}^{\Vert \cdot {\Vert }_{{\mathcal{E}}_D}}={\overline{C_c^{\infty }(D)}}^{\Vert \cdot {\Vert }_{\mathcal{E}}}.$$

$${\mathcal{E}}_D(u,u)=c_{\alpha ,d}[u{]}_{H^{\alpha }(D)}+c_{\alpha ,d}{\int }_D{u}^2(x)c(x)dx,$$

$$c(x)=2{\int }_{{\mathbb{R}}^d\setminus D}\frac{1}{|x-y{|}^{d+2\alpha }}dy,x\in D.$$

$${\mathcal{E}}_D(u,u)=\mathcal{E}(u,u)\le C_3\Vert u{\Vert }_{H^{\alpha }({\mathbb{R}}^d)}^2\le C_4\Vert u{\Vert }_{H^{\alpha }(D)}^2.$$

$$c_1\Vert u{\Vert }_{{\mathcal{E}}_D}\le \Vert u{\Vert }_{H^{\alpha }(D)}\le c_2\Vert u{\Vert }_{{\mathcal{E}}_D},u\in D({\mathcal{E}}_D)=H_0^{\alpha }(D).$$

$$H^{\alpha }(D)\subset L^p(D;m),$$

Full text

††AMS subject classifications. Primary 35B40; Secondary 35K57, 35J61, 35R11, 35K85.

Keywords. Fractional Laplace operator, logistic equation, asymptotics, obstacle problem, Feynman-Kac formula, Lévy process, intrinsic ultracontractivity.

1. Introduction

Let $D\subset{\mathbb{R}}^{d}$ ($d\geq 2$) be a bounded Lipschitz domain, $\varphi,b$ be bounded positive Borel measurable functions on $D$ and $a>0$. In the present paper, we investigate asymptotics, as $p\rightarrow\infty$ and $t\rightarrow\infty$, of solutions to the following Cauchy-Dirichlet problem

[TABLE]

where $\alpha\in(0,1)$, and $\Delta^{\alpha}$ is the fractional Laplacian (see Section 2 for details). Equations and systems of type (1.1) serve as basic models in population biology. In classical models, $\alpha=1$, the operator involved in (1.1) is the usual Laplace operator. In the present paper, we concentrate on the study of (1.1) with nonlocal operators, $\alpha\in(0,1)$. In recent years, nonlocal population models attracted quite a lot interest (see [1, 9, 36, 38, 49] and the references therein). They are designed to describe the nonlocal dispersal strategy of animals. This type of dispersal strategy based on Lévy flights has been observed in nature (see, e.g., [24, 36] for a discussion of this problem). Very recently, Caffarelli, Dipierro and Valdinoci [8] investigated the existence problem for steady-state population model of type (1.1) with additional nonlocal term on the right-hand side describing the nonlocal character of the species rate.

In the case of the classical Laplace operator, Dancer and Du [11, 12] proved a very interesting result stating that for large $p\geq 1$ the solutions of a stationary counterpart to (1.1) behave like solutions of certain steady-state predator-pray models. This common behaviour was described by certain free boundary problem.

In the present paper, motivated by the results of Dancer and Du, we study the asymptotic behaviour of solutions to (1.1). We consider the following two cases:

1. (i)

we pass to the limit in (1.1) with $p\rightarrow+\infty$ and then with $t\rightarrow+\infty$; 2. (ii)

we pass to the limit in (1.1) with $t\rightarrow+\infty$ and then with $p\rightarrow+\infty$.

The most interesting part is the convergence as $p\rightarrow+\infty$ because by the known results for the usual Laplace operator (see [6, 7, 10, 14, 43]), it is reasonable to expect that the limit function is a solution of some free boundary problem (or, equivalently, the obstacle problem). This phenomenon was studied for the first time by Boccardo and Murat [6] in the case of equations with Leray-Lions type operator and with $a=0$, $b=1$. An interesting part is also the convergence as $t\rightarrow\infty$ in (i). It implies the large-time asymptotics for an evolution obstacle problem and, at the same time, an existence result for a stationary obstacle problem. Asymptotics of solutions to equations of type (1.1) with classical Laplacian and general $a,b$ was investigated in [14, 43]. To our knowledge, there are no asymptotics results for (1.1) with $\alpha\in(0,1)$ when $p\rightarrow\infty$ in (i) and (ii), and in (i) when $t\rightarrow\infty$.

In the whole paper, we assume that the following hypotheses are satisfied.

1. (H1)

$b\in{\cal B}_{b}^{+}(D)$ is nontrivial (i.e. $\int_{D}b\,dm>0$, where $m$ is the Lebesgue measure), there exists a Lipschitz domain $D_{0}\subset D$ such that $\{b=0\}=\overline{D}_{0}$, and for every compact $K\subset D\setminus\overline{D}_{0}$,

[TABLE] 2. (H2)

$\varphi\in{\cal B}^{+}_{b}(D)$ is nontrivial and $0\leq\varphi\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$, where (with the convention $\infty\cdot 0=0$)

[TABLE]

One of the main difficulty in studies on equations of type (1.1) lies in the fact that $b$ may vanish (the so called degenerate logistic equations). When $b$ is bounded away from zero, then the term $bv_{p}^{p}$ in (1.1) is bounded (uniformly in $p\geq 1$) in $L^{q}$ norm for any $q\geq 1$, however, if we assume that $b$ is merely non-trivial, then we are losing some control on the term $bv_{p}^{p}$, and the best we can get in the limit (as $p\to\infty$) is a bounded measure. The techniques proposed and developed by Dancer, Du and Ma in [11, 12, 13, 14], and Rodrigues and Tavares in [43] strongly exploits the properties inherent to the Laplace operator as locality of the operator - the evaluation of $\Delta u(x)$ depends on the values of $u$ in an arbitrarily small neighborhood of $x$ - and regularity up to the boundary of solutions to Poisson equations with smooth data and domains. Unfortunately, the fractional Laplacian does not share these properties (see [44]). Therefore, we find ourselves forced to propose a new method of studying (1.1). It combines the techniques used in the case of the classical Laplacian with some new technics based on the probabilistic potential theory and stochastic analysis. Considered method allows us to handle asymptotics for (1.1) with irregular data and domain, and also to get pointwise convergence in asymptotics results. The last property was never investigated in the literature in the case of degenerate logistic type equations. The method we apply here depends upon the knowledge that $\Delta^{\alpha}$ is symmetric, strongly Feller and intrinsic ultracontractive, therefore the results of the paper may be easily extended to a much broader class of operators (including classical Laplacian). The technique seems to be very powerful as evidenced by the fact that when applied to classical Laplacian, it gives stronger results than in [11, 12, 14, 43] (this is so, among others, because we do not use the Hopf lemma, which requires high regularity of the boundary of $D$). Note that in the paper [43] devoted to evolution equations, the authors assume in addition that $D,D_{0}$ are smooth domains, and in [14] devoted to elliptic equations the authors assume in addition that $D,D_{0}$ are smooth, $b$ is continuous and $\overline{D}_{0}\subset D$.

As for (i), we prove that if $v$ is a unique solution of the parabolic obstacle problem

[TABLE]

then for all $T>0$ and $\delta\in(0,T]$,

[TABLE]

Moreover, if $\varphi\in C_{0}(D)$, then (1.3) holds with $\delta=0$. We then show that for every $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$, where $\lambda_{1}^{D}$ (resp. $\lambda_{1}^{D_{0}}$) denotes the principal eigenvalue of the operator $-\Delta^{\alpha}$ with zero exterior Dirichlet condition on $D^{c}$ (resp. $D_{0}^{c}$), there exists a (unique) solution $u$ of the elliptic obstacle problem

[TABLE]

and

[TABLE]

As a matter of fact, in the present paper, we only show the existence of $u$. The uniqueness problem for (1.4) is a separate difficult issue. It is solved in [30] (see also [13] for the case of the classical Laplacian).

As for problem (ii), we show that there exists a solution $u_{p}$ of the problem

[TABLE]

if and only if $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$, and that for $a$ satisfying this condition,

[TABLE]

We next show that for every $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$,

[TABLE]

where $u$ is a solution to (1.4). The uniform convergence in (1.3) and (1.8) has been considered before in the literature only in the case when $b\geq c$ for some constant $c>0$. For the proof in the general case, we combine the analytic methods of [6, 43] with the Feynman-Kac representation and some methods of stochastic analysis and probabilistic potential theory. In the proof of the asymptotics as $t\rightarrow\infty$, we merge the techniques introduced in [45] with the probabilistic ones introduced in [25]. Note here that under additional regularity conditions on $b,D,D_{0},\varphi$ the large time behaviour of solutions to (1.1) and (1.2) with classical Laplace operator was studied in [18, 20, 21, 43].

The method we propose in the paper is built on three pillars: the Feynman-Kac representation of solutions to the mentioned problems (thanks to which, among others, we achieve uniform convergences), the notion of intrinsic ultracontractivity which stands as a substitute of the Hopf lemma (as a byproduct, we may consider less regular domains), and the following result which plays a pivotal role in our proofs of the energy estimates for solutions to (1.1):

[TABLE]

Here $V\subset D$ is a bounded Kac regular domain (any Lipschitz domain is Kac regular, see Proposition 2.2) and $\lambda^{V}(-\Delta^{\alpha}+q_{k})$ is the principal eigenvalue of the operator $-\Delta^{\alpha}+q_{k}$ with zero exterior Dirichlet condition on $V^{c}$. Furthermore, $\{q_{k}\}$ is an increasing sequence of bounded positive measurable functions on $D$ such that supp$[q_{k}]\subset D\setminus\overline{V}$ and

[TABLE]

Similar result, but for classical Dirichlet Laplacian, was proved in [21] under very restrictive smoothness assumptions on the domain.

It is worth mentioning that there is a rich series of papers (see [1, 9] and the references therein) devoted to nonlocal logistic equations of the form (1.1) but with $\Delta^{\alpha}$ replaced by a nonlocal operator $A$ of the form

[TABLE]

with some strictly positive symmetric kernel $J\in C(\overline{D}\times\overline{D})$. By the very definition of the fractional Laplacian representation (1.10) holds for $A=\Delta^{\alpha}$ (with the principal value integral) but for $J(x,y)\sim|x-y|^{-d-2\alpha}$, which clearly does not belong to $C(\overline{D}\times\overline{D})$.

2. Preliminary results

Throughout the paper, we assume that $d\geq 2$ and $D\subset{\mathbb{R}}^{d}$ is a bounded Lipschitz domain. We let $m$ denote the Lebesgue measure on ${\mathbb{R}}^{d}$. We denote by ${\cal B}(D)$ the $\sigma$-field of Borel subsets of $D$. ${\cal B}_{b}(D)$ (resp. ${\cal B}^{+}(D)$) is the set of real bounded (resp. positive) Borel measurable functions on $D$. ${\cal B}^{+}_{b}(D)={\cal B}_{b}(D)\cap{\cal B}^{+}(D)$.

2.1. Dirichlet fractional Laplacian and related Sobolev spaces

For any $u\in C^{2}_{b}({\mathbb{R}}^{d})$ we let

[TABLE]

with $c_{\alpha,d}=[4^{\alpha}\Gamma(\frac{d+2\alpha}{2})]/[\pi^{d/2}\Gamma(-\alpha)]$.

Let us consider the Dirichlet form $({\cal E},D({\cal E}))$ on $L^{2}({\mathbb{R}}^{d};m)$ defined as

[TABLE]

Here $\hat{u}$ stands for the Fourier transform of $u$. By [19, Proposition 3.4, Theorem 6.5] (see also [37, Lemma 3.15]) $D({\cal E})=H^{\alpha}({\mathbb{R}}^{d})$, and there exist $C_{1},C_{2}>0$ such that $C_{1}\|u\|_{{\cal E}}\leq\|u\|_{H^{\alpha}({\mathbb{R}}^{d})}\leq C_{2}\|u\|_{{\cal E}},\,u\in D({\cal E})$, where $\|u\|_{\cal E}=\sqrt{{\cal E}(u,u)}$, and

[TABLE]

with

[TABLE]

Moreover, by [19, Proposition 3.3]

[TABLE]

By [22, Sections 1.3, 1.4] there exists a unique self-adjoint operator $(A,D(A))$ such that $D(A)\subset D({\cal E})$, and

[TABLE]

From this relation and (2.4) we infer that $C^{2}_{c}({\mathbb{R}}^{d})\subset D(A)$, and for any $u\in C_{c}^{2}({\mathbb{R}}^{d})$

[TABLE]

Let Cap be the capacity naturally associated with the form ${\cal E}$ (see [22, Section 2.1]). We say that a property holds ${\cal E}$-q.e. if it holds outside a set of capacity Cap zero. We say that a function $u$ on ${\mathbb{R}}^{d}$ is ${\cal E}$-quasi-continuous if for every $\varepsilon>0$ there exists a closed set $F_{\varepsilon}$ such that $\mbox{Cap}({\mathbb{R}}^{d}\setminus F_{\varepsilon})\leq\varepsilon$ and $u_{|F_{\varepsilon}}$ is continuous. It is well known (see [22, Theorem 2.1.3]) that each $u\in D({\cal E})$ has an ${\cal E}$-quasi-continuous $m$-version, which in the sequel will be denoted by $\tilde{u}$.

By $\{T_{t},\,t\geq 0\}$ (resp. $\{J_{\beta},\,\beta\in\rho(A)\}$), we denote the semigroup (resp. resolvent) generated by $A$.

We let $({\cal E}_{D},D({\cal E}_{D}))$ denote a form (called the part of $({\cal E},D({\cal E}))$ on $D$) defined by

[TABLE]

By [22, Theorem 4.4.3], $({\cal E}_{D},D({\cal E}_{D}))$ is a regular Dirichlet form on $L^{2}(D;m)$. Therefore (see [22, Sections 1.3, 1.4]) there exists a unique self-adjoint negative definite operator $(A_{D},D(A_{D})$ on $L^{2}(D;m)$ such that

[TABLE]

(here $(\cdot,\cdot)$ stands for the usual scalar product in $L^{2}({\mathbb{R}}^{d};m)$). We put

[TABLE]

The operator $(\Delta^{\alpha})_{|D}$ is called the Dirichlet fractional Laplacian. Let $H^{\alpha}(D),[u]_{H^{\alpha}(D)}$ be defined as in (2.2), (2.3) but with ${\mathbb{R}}^{d}$ replaced by $D$. Let

[TABLE]

The last equation follows from [19, Lemma 5.1]. On the other hand, since $({\cal E}_{D},D({\cal E}_{D}))$ is regular, then

[TABLE]

The last equation follows from the definition of the form $({\cal E}_{D},D({\cal E}_{D}))$. By the equivalence of the norms $\|\cdot\|_{{\cal E}}$ and $\|\cdot\|_{H^{\alpha}({\mathbb{R}}^{d})}$, we conclude that $D({\cal E}_{D})=H^{\alpha}_{0}(D)$. Observe that by (2.4) for any $u\in D({\cal E}_{D})$,

[TABLE]

where

[TABLE]

Thus, by [19, Theorem 6.7], there exists $c_{1}>0$ such that $c_{1}\|u\|^{2}_{H^{\alpha}(D)}\leq{\cal E}_{D}(u,u),\,u\in D({\cal E}_{D})$. On the other hand for any $u\in C_{c}^{\infty}(D)$, by the equivalence of the norms $\|\cdot\|_{{\cal E}}$ and $\|\cdot\|_{H^{\alpha}({\mathbb{R}}^{d})}$, and [19, Theorem 5.4], we have

[TABLE]

Consequently, there exist $c_{1},c_{2}>0$ such that

[TABLE]

Recall here (see e.g. [19, Corollary 7.2]) that for any $p\in[1,2d/(d-2\alpha))$,

[TABLE]

and the embedding is compact.

As in the case of the form $({\cal E},D({\cal E}))$, one can define capacity $\mbox{Cap}_{D}$ and the notions of ${\cal E}_{D}$-exceptional sets and ${\cal E}_{D}$-quasi-continuity. We will drop ${\cal E}_{D}$ in the notation if it will be clear from the context which Dirichlet form is considered. Note, however, that on the set $D$ both capacities, i.e. $\mbox{Cap}_{D}$ and Cap are equivalent, and the notions of ${\cal E}_{D}$-quasi-continuity and ${\cal E}$-quasi-continuity agree (see [22, Theorem 4.4.3]).

2.2. Probabilistic potential theory

Let $\mathbb{X}=((X_{t})_{t\geq 0},(P_{x})_{x\in{\mathbb{R}}^{d}},({\cal F}_{t})_{t\geq 0})$ be a rotation invariant $\alpha$-stable Lévy process associated with $({\cal E},D({\cal E}))$ in the sense that for any positive Borel function $f\in L^{2}(E;m)$,

[TABLE]

where $\mathbb{E}_{x}$ denotes the expectation with respect to the measure $P_{x}$. It is well known (see e.g. [2, Proposition I.2.5] and [2, Exercise 4, page 39]) that such a process is doubly Feller, i.e. it is strongly Feller: $P_{t}({\cal B}_{b}({\mathbb{R}}^{d}))\subset C_{b}({\mathbb{R}}^{d}),\,t>0$, and it is Fellerian: $P_{t}(C_{0}({\mathbb{R}}^{d}))\subset C_{0}({\mathbb{R}}^{d}),\,t>0$. Here $C_{b}({\mathbb{R}}^{d})$ is the set of bounded continuous functions on ${\mathbb{R}}$, and $C_{0}({\mathbb{R}}^{d})$ is the set of continuous functions on ${\mathbb{R}}^{d}$ vanishing at infinity. We denote by $\mathbb{X}^{D}$ the process $\mathbb{X}$ killed upon exiting $D$. It is known that ${\mathbb{X}}^{D}$ is associated with the form $({\cal E}_{D},D({\cal E}_{D}))$ (see [22, Theorem 4.4.2]). This means that for any positive Borel $f\in L^{2}(D;m)$,

[TABLE]

[TABLE]

where

[TABLE]

Here $(T^{D}_{t})$ is a Markov semigroup generated by $(\Delta^{\alpha})_{|D}$ on $L^{2}(D;m)$, and $(J^{D}_{\beta})$ is its resolvent (note that $[0,\infty)$ is included in the resolvent set of $(\Delta^{\alpha})_{|D}$). We denote by $G_{D,\beta}$ the $\beta$-Green function for the operator $-(\Delta^{\alpha})_{|D}$, and by $p_{D}$ its transition function (see [22, Exercise 4.2.1, Lemma 4.2.4] for details). By the definition, for any $f\in{\cal B}^{+}(D)$,

[TABLE]

We write $G_{D}=G_{D,0}$. For a positive Borel measure $\mu$ on $D$, we set

[TABLE]

and $R^{D}=R^{D}_{0}$. By [22, Theorem 4.2.3], for any $f\in{\cal B}(D)\cap L^{2}(D;m)$ we have

[TABLE]

It is well known (see e.g. [23, Lemma 2.1]), that

[TABLE]

It is also well known (see e.g. [33, Section 4.2]) that there exists $c>0$ (depending on $\alpha,d$) such that

[TABLE]

This in turn implies that

[TABLE]

We say that a Borel measure $\mu$ on $D$ is ${\cal E}_{D}$-smooth if $|\mu|\ll\mbox{Cap}_{D}$ (i.e. $\mu$ charges no set of capacity $\mbox{Cap}_{D}$ zero) and there exists an increasing sequence $\{F_{n}\}$ of closed subsets of $D$ such that $|\mu|(F_{n})<\infty$, $n\geq 1$, and $\mbox{Cap}_{D}(K\setminus F_{n})\rightarrow 0$ as $n\rightarrow\infty$ for every compact set $K\subset D$.

We let ${\cal M}_{0,b}(D)$ be the space of all bounded ${\cal E}_{D}$-smooth measures on $D$. We say that an ${\cal E}_{D}$-smooth measure $\mu$ belongs to the class $S_{0}(D)$ if there exists $c>0$ such that

[TABLE]

In the light of (2.5), $S_{0}(D)\subset H^{-\alpha}(D)$. We denote by $\|\mu\|_{H^{-\alpha}(D)}$ the smallest $c\geq 0$ such that (2.14) holds. It is well known that there is a one-to-one correspondence (Revuz duality) between positive ${\cal E}_{D}$-smooth measures and positive continuous additive functionals (PCAFs for short) of $\mathbb{X}^{D}$ (see [22, Theorem 5.1.4]). By [22, Theorem 5.1.3], if $A^{\mu}$ is the unique PCAF associated with a positive smooth measure $\mu$, then for every positive Borel function $f$ on $D$,

[TABLE]

where $f\cdot\mu$ is a positive Borel measure such that $d(f\cdot\mu)/d\mu=f$.

In what follows, if there is no ambiguity, in the notation we drop the prefix ${\cal E}_{D}$.

Lemma 2.1**.**

Assume that $u\in{\cal B}(D)$ and $\mu\in\mathcal{M}_{0,b}(D)$. Then

[TABLE]

for q.e. $x\in D$ if and only if there exists a process $M$ with $M_{0}=0$ such that $M$ is a uniformly integrable martingale on $[0,\tau_{D}]$ under the measure $P_{x}$ for q.e. $x\in{\mathbb{R}}^{d}$, and

[TABLE]

Proof.

See [31, Theorem 4.7]. ∎

2.3. Regular domains

We say that a bounded domain $D\subset{\mathbb{R}}^{d}$ is Dirichlet regular if for every $x\in{\mathbb{R}}^{d}\setminus D$,

[TABLE]

To see that any Lipschitz domain is Dirichlet regular, we first recall the notion of the base of a set $A\subset{\cal B}({\mathbb{R}}^{d})$ (see [3, Section VI.4]). Let

[TABLE]

We have that for any $A,B\subset{\cal B}({\mathbb{R}}^{d})$,

[TABLE]

Indeed, the inclusion $b(A)\cup b(B)\subset b(A\cup B)$ is clear (since $\sigma_{A\cup B}\leq\sigma_{A}\wedge\sigma_{B}$). For the reverse inclusion, observe that $\sigma_{A\cup B}=0$ implies that $\sigma_{A}=0$ or $\sigma_{B}=0$. Therefore, if $x\in b(A\cup B)$, then $P_{x}(\sigma_{A}=0)>0$ or $P_{x}(\sigma_{B}=0)>0$. By Blumenthal’s zero–one law this implies that $P_{x}(\sigma_{A}=0)=1$ or $P_{x}(\sigma_{B}=0)=1$. Thus, $x\in b(A)\cup b(B)$.

The following reasoning is taken from [3, Example VI.4.7.4, page 276]. Let $x\in\partial D$, and $C_{x}$ be an open exterior cone at $x$. There exist rotations $\Phi_{1},\dots\Phi_{m}:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ such that $\overline{D}^{c}\setminus\{x\}=\bigcup_{i=1}^{m}\Phi_{i}(\overline{D}^{c}\cap C_{x})$. Clearly, $x\in b(\overline{D}^{c}\setminus\{x\})$, so by (2.16) there exists $i_{0}\in\{1,\dots,m\}$ such that $x\in b(\Phi_{i_{0}}(\overline{D}^{c}\cap C_{x}))$. By the rotation invariance of $\mathbb{X}$, $x\in b(\overline{D}^{c}\cap C_{x})$. Thus, $P_{x}(\sigma_{\overline{D}^{c}\cap C_{x}}=0)=1$. Consequently,

[TABLE]

Hence, $P_{x}(\tau_{D}>0)=0,\,x\in D^{c}$. We see that exterior cone condition is sufficient for Dirichlet regularity of $D$. In particular Lipschitz domains are Dirichlet regular.

We say that a domain $D$ is Kac regular if

[TABLE]

Proposition 2.2**.**

If $D$ is a bounded Lipschitz domain, then $D$ is Kac regular.

Proof.

Let $x\in D$. Clearly, $\tau_{D}\leq\tau_{\overline{D}}$. By the strong Markov property

[TABLE]

By (2.17), $P_{y}(\tau_{\overline{D}}=0)=1,\,y\in\partial D$. If $y\in\overline{D}^{c}$, then obviously $P_{y}(\tau_{\overline{D}}=0)=1$. Thus,

[TABLE]

From this and (2.19), we get (2.18). ∎

2.4. Eigenfunctions and intrinsic ultracontractivity

In what follows, we denote by $\lambda_{1}(B)$ the first eigenvalue (whenever it exists) of a given operator $B$. To simplify notation, we also set

[TABLE]

where $q:D\to{\mathbb{R}}^{+}$ is a positive potential. It is well known (see e.g. [4]) that

[TABLE]

Therefore, by Jentzsch’s theorem (see e.g. [46, Theorem V.6.6, page 338] or [22, Lemma 6.4.5]) for any bounded open domain $D\subset{\mathbb{R}}^{d}$ and a positive $q\in{\cal B}_{b}(D)$ there exists a unique strictly positive continuous eigenfunction $\psi$ associated with the eigenvalue $\lambda^{D}_{1}[q]>0$ such that $\|\psi\|_{L^{2}(D;m)}=1$. We call $\psi$ the principal eigenfunction for the operator $-(\Delta^{\alpha})_{|D}+q$, and $\lambda_{1}^{D}[q]$ the principal eigenvalue. Moreover, if $D$ is Dirichlet regular, then $\psi\in C_{0}(D)$. We denote by $\varphi_{1}^{D}$ the principal eigenfunction for $-(\Delta^{\alpha})_{|D}$. From (2.20) it also follows that for any $q\in[1,(d+2\alpha)/d)$ and $T>0$,

[TABLE]

Recall that a symmetric Markov semigroup $(Q_{t})$ on $L^{2}(D;m)$ is said to be ultracontractive if for any $t>0$, $Q_{t}:L^{2}(D;m)\to L^{\infty}(D;m)$ is bounded. In this case, there exists a transition function $q(\cdot,\cdot,\cdot)$ such that for any $f\in{\cal B}^{+}(D)$,

[TABLE]

and for any $t>0$ there exists $c_{t}>0$ such that $q(t,x,y)\leq c_{t},\,x,y\in D$. By [23], the semigroup $(P^{D}_{t})_{t>0}$ is intrinsically ultracontractive (the notion introduced in [16]), i.e. for any $t>0$, a Markov semigroup $(Q^{D}_{t})$ defined as

[TABLE]

is ultracontractive on $L^{2}(D,\nu)$ with $\nu:=(\varphi_{1}^{D})^{2}\cdot m$. Observe that the transition density $q_{D}(\cdot,\cdot,\cdot)$ for $(Q^{D}_{t})$ admits the following formula

[TABLE]

Therefore, for any $t>0$, there exists $\beta_{t}>0$ such that

[TABLE]

By [15, Lemma 2.1.2], $\beta_{t}$ is non-increasing as $t\rightarrow\infty$. Moreover, by [16, Theorem 3.2] for any $t>0$, there exists $\alpha_{t}>0$ such that

[TABLE]

3. Eigenfunctions and eigenvalues

From now on, unless it is stated otherwise, we assume that (H1), (H2) (see Introduction) are satisfied. Recall that by the Rayleigh-Ritz variational formula (see e.g. [15, Theorem 4.5.1]),

[TABLE]

for any bounded open domain $D\subset{\mathbb{R}}^{d}$, and for any $q\in{\cal B}_{b}^{+}(D)$. In particular, for open bounded domains $D_{0}\subset D\subset{\mathbb{R}}^{d}$, we have

[TABLE]

Theorem 3.1**.**

Let $c\in{\cal B}^{+}_{b}(D)$, and $\{q_{k}\}\subset{\cal B}^{+}_{b}(D)$ be a sequence such that supp$[q_{k}]\subset D\setminus\overline{D}_{0}$, $k\geq 1$, and for every compact $K\subset D\setminus\overline{D}_{0}$, $\inf_{x\in K}q_{k}(x)\nearrow\infty$ as $k\rightarrow\infty$. Let $\psi_{k}^{c}$ (resp. $\psi^{c}$) be the principal eigenfunction for $-(\Delta^{\alpha})_{|D}+c+q_{k}$ (resp. $-(\Delta^{\alpha})_{|D}+c$). Then

[TABLE]

and there exists a subsequence $\{k_{n}\}$ such that

[TABLE]

Proof.

By the definition of the principal eigenvalue,

[TABLE]

By [22, Theorem 6.1.1],

[TABLE]

By (3.1),

[TABLE]

From this and (3.2), we get

[TABLE]

By (2.6), there exists a subsequence $\{n_{k}\}$ such that $\psi^{c}_{k_{n}}\rightarrow\psi$ in $L^{2}(D;m)$ for some $\psi\in L^{2}(D;m)$ such that $\|\psi\|_{L^{2}(D;m)}=1$. Let $\nu\in S_{0}(D)$ (cf. (2.14)) be a positive measure such that $R^{D}\nu$ is bounded q.e by a constant. Then

[TABLE]

From this and a standard reasoning (see the reasoning following [22, (5.2.22)]), we infer that, up to subsequence,

[TABLE]

Applying this convergence and (3.3) we deduce that, up to a subsequence, $\{\psi^{c}_{k_{n}}\}$ is convergent q.e. Set $\tilde{\psi}(x)=\limsup_{n\rightarrow\infty}\psi^{c}_{k_{n}}(x)$, $x\in D$. Observe that by the assumptions on the sequence $\{q_{k}\}$,

[TABLE]

with the convention that $0\cdot\infty=0$. Hence

[TABLE]

From this, (3.3), (3.5) and q.e. convergence of $\psi^{c}_{k_{n}}$ we conclude

[TABLE]

By Proposition 2.2,

[TABLE]

By the above formula, we get, in particular, that $\tilde{\psi}$ is strictly positive and quasi-continuous on $D_{0}$. Since $\tilde{\psi}=\psi$ a.e., $\tilde{\psi}\in D({\cal E}_{D})$. Hence, by [22, Theorem 6.1.1], $\tilde{\psi}$ is a strictly positive solution to $-(\Delta^{\alpha})_{|D_{0}}u+cu=\lambda u$. Therefore $\tilde{\psi}=\psi^{c}$ q.e. and $\lambda=\lambda^{D_{0}}_{1}[c]$. ∎

Lemma 3.2**.**

Let $q_{1},q_{2}\in{\cal B}^{+}_{b}(D)$. If $q_{1}\leq q_{2}$ a.e. and $m(\{q_{1}<q_{2}\})>0$, then

[TABLE]

Proof.

Let $\psi_{1},\psi_{2}$ be the principal eigenfunctions for $\lambda^{D}_{1}[q_{1}]$ and $\lambda^{D}_{1}[q_{2}]$, respectively. It is clear that $\lambda^{D}_{1}[q_{1}]\leq\lambda^{D}_{1}[q_{2}]$. Suppose that $\lambda:=\lambda^{D}_{1}[q_{1}]=\lambda^{D}_{1}[q_{2}]$. Then

[TABLE]

Hence $(q_{2}-q_{1},\psi_{1}\psi_{2})=0$, which contradicts the fact that $m(\{q_{1}<q_{2}\})>0$. ∎

Lemma 3.3**.**

We have $\lambda^{D}_{1}<\lambda^{D_{0}}_{1}$ (cf. (H1)).

Proof.

Let $\{q_{k}\}\subset{\cal B}^{+}_{b}(D)$ be a sequence of functions such that supp$[q_{k}]\subset D\setminus\overline{D}_{0}$ and for every compact $K\subset D\setminus\overline{D}_{0}$, $\inf_{x\in K}q_{k}(x)\nearrow\infty$. By Theorem 3.1, $\lambda_{1}^{D}[q_{k}]\nearrow\lambda_{1}^{D_{0}}$ as $k\rightarrow\infty$, whereas by Lemma 3.2, $\lambda_{1}^{D}<\lambda_{1}^{D}[q_{k}]$, $k\geq 1$, which proves the lemma. ∎

4. Existence result for semilinear elliptic equations

We recall that we assume that (H1), (H2) are satisfied and $D$ is a bounded Lipschitz domain.

Let $f:D\times\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function which is bounded on bounded subsets of $D\times{\mathbb{R}}$. We consider the following problem:

[TABLE]

Definition 4.1**.**

We say that a bounded function $u$ on $D$ is a solution to (4.1) if for q.e. $x\in D$,

[TABLE]

Remark 4.2**.**

In the present section, we choose as basis the above integral form definition of a solutions to PDE (4.1) since it is suitable for the method of sub and supersolutions we apply below. However, the above definition is related to the following more familiar definitions. Let $u$ be a bounded function on ${\mathbb{R}}^{d}$.

• a)

By (2.10), $u$ is a solution to (4.1) if and only if $u$ is a weak solution to (4.1), i.e. $u\in H^{\alpha}_{0}(D)$ and

[TABLE]

• b)

By (2.8), $u$ is a solution to (4.1) if and only if $u\in D(-(\Delta^{\alpha})_{|D})$, and

[TABLE]

in $L^{2}(D;m)$.

• c)

By [33, Theorem 7.1], if $u$ is a continuous solution to (4.1), then

[TABLE]

i.e. the limit in (2.1) exists and the above equation holds for any $x\in{\mathbb{R}}^{d}\setminus D$. Conversely, assume that $u\in{\cal B}_{b}({\mathbb{R}}^{d})\cap C(D)$ and satisfies the above equations, i.e. $u$ is a pointwise solution to (4.1). Then $u$ is a solution to (4.1) (see e.g. [33, Theorem 7.2]).

Definition 4.3**.**

We say that a bounded function $u$ on $D$ is a supersolution (resp. subsolution) to (4.1) if there exists a positive (resp. negative) measure $\mu\in{\cal M}_{0}(D)$ such that for q.e. $x\in D$,

[TABLE]

Proposition 4.4**.**

Let $\underline{u}$ (resp. $\bar{u}$) be a subsolution (resp. supersolution) to (4.1) and $\underline{u}\leq\bar{u}$. Then there exists a solution $u$ to (4.1) such that $\underline{u}\leq u\leq\bar{u}$.

Proof.

Step 1. Define

[TABLE]

We shall show that if $\hat{u}$ is a solution to (4.1), with $f$ replaced by $\hat{f}$, then $\underline{u}\leq\hat{u}\leq\bar{u}$. By the definition of a supersolution, there exists a positive $\mu\in{\cal M}_{0,b}(D)$ such that

[TABLE]

By Lemma 2.1, there exist martingales $\hat{M},\bar{M}$ such that for q.e. $x\in D$,

[TABLE]

[TABLE]

By the Tanaka-Meyer formula (see, e.g., [42, IV.Theorem 70]),

[TABLE]

By the definition of $\hat{f}$ and positivity of $\mu$ we get $(\hat{u}-\bar{u})^{+}=0$. A similar argument shows that $(\underline{u}-\hat{u})^{+}=0$. Thus $\underline{u}\leq\hat{u}\leq\bar{u}$ as claimed.

Step 2. Define $\Phi$ by

[TABLE]

Since $\hat{f}$ is bounded, the operator $\Phi$ is well defined (cf. (2.11)). From continuity of $f$ it follows at once that $\Phi$ is continuous. Let $\{u_{n}\}\subset L^{2}(D;m)$. By (2.6), (2.10), there exists a subsequence $\{n_{k}\}$ such that $R^{D}\hat{f}(\cdot,u_{n_{k}})$ is convergent a.e. Applying (2.11) and the dominated convergence theorem shows the convergence of $\{R^{D}\hat{f}(\cdot,u_{n})\}$ in $L^{2}(D;m)$. Thus $\Phi$ is compact. Therefore, by Schauder’s fixed point theorem, there exists $u\in L^{2}(D;m)$ such that $u=R^{D}\hat{f}(\cdot,u)$ a.e. Of course, we may choose an $m$-version $\hat{u}$ of $u$ such that $\hat{u}=R^{D}\hat{f}(\cdot,\hat{u})$. By Step 1, $\underline{u}\leq\hat{u}\leq\bar{u}$, so $\hat{u}=R^{D}f(\cdot,\hat{u})$. ∎

Proposition 4.5**.**

Assume that $a>0$. Then there exists at most one strictly positive solution to (1.6).

Proof.

Let $u_{1},u_{2}$ be strictly positive solutions to (1.6). It is an elementary check that $u_{1}+u_{2}$ is a supersolution to (1.6). It is also well known (see e.g. [30, Proposition 3.7]) that $u_{1}\vee u_{2}$ is a subsolution to (1.6). This, when combined with Proposition 4.4, shows that without loss of generality we may assume that $u_{1}\leq u_{2}$. Striving for a contradiction, suppose that $m(\{u_{1}<u_{2}\})>0$. By the Feynman-Kac formula (see e.g. [22, Theorem 6.1.1]), for every $x\in D$,

[TABLE]

with $\beta=\|b\|_{\infty}\|u_{2}\|^{p-1}$. It follows from this and (2.20) that $u_{1}(x)<u_{2}(x),\,x\in D$. In particular, since $b$ is nontrivial and positive, $m(\{b(u_{1})^{p-1}<b(u_{2})^{p-1}\})>0$. Hence, by Lemma 3.2,

[TABLE]

which is a contradiction. ∎

Theorem 4.6**.**

There exists a solution to (1.6) if and only if $\lambda_{1}^{D}<a<\lambda_{1}^{D_{0}}$.

Proof.

Suppose that there exists a solution $u$ to (1.6). Then, by Lemma 3.2,

[TABLE]

Moreover, by Lemma 3.3 and (H1),

[TABLE]

Now, assume that $\lambda_{1}^{D}<a<\lambda_{1}^{D_{0}}$. By Theorem 3.1 and the fact that $a<\lambda_{1}^{D_{0}}$, there exists a positive $\eta\in C_{c}^{\infty}(D)$ with supp$[\eta]\subset D\setminus\overline{D}_{0}$ such that $\lambda_{1}^{D}[\eta]\geq a$. By (H1), there exists a positive function $v\in C^{\infty}_{c}(D)$ such that $bv^{p-1}\geq\eta$. We thus have that $\lambda_{1}^{D}[bv^{p-1}]\geq\lambda_{1}^{D}[\eta]\geq a$. Let $\psi$ be the principal eigenfunction for $\lambda_{1}^{D}[bv^{p-1}]$, and let $c>0$ be such that $c\psi\geq v$. Then

[TABLE]

Therefore $c\psi$ is a supersolution to (1.6). It is clear that for a sufficiently small $\varepsilon>0$, $\varepsilon\varphi_{1}^{D}$ is a subsolution to (1.6). Moreover, by the Feynman-Kac formula (see e.g. [22, Theorem 6.1.1]) and ultracontractivity of $p_{D}$, for every $x\in D$ we have

[TABLE]

Thus, for a sufficiently small $\varepsilon>0$, $\varepsilon\varphi_{1}^{D}\leq c\psi$. Therefore, by Proposition 4.4, there exists a solution to (1.6). ∎

5. Obstacle problem and asymptotics as $p\rightarrow\infty$ for elliptic equations

In this section, we provide three equivalent formulations of the obstacle problem (1.4). All three shall prove to be useful throughout the paper. Next, we prove asymptotics of steady-state logistic equations with respect to the increasing power of the absorption term. As a by-product, we get an existence result for the obstacle problem (1.4). As in Sections 3 and 4, we assume that (H1), (H2) are in force and $D$ is a bounded Lipschitz domain.

5.1. Obstacle problem

Definition 5.1**.**

We say that $u\in H^{\alpha}_{0}(D)$ is a weak solution to (1.4) if $0<u\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. and for every $\eta\in H^{\alpha}_{0}(D)$ such that $\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. we have

[TABLE]

Proposition 5.2**.**

Assume that $u$ is a quasi-continuous bounded strictly positive function on $D$ such that $u\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. Then the following statements are equivalent:

1. (i)

$u$* is a weak solution to (1.4).* 2. (ii)

There exists a positive $\mu\in S_{0}(D)$ such that

1. (a)

${\cal E}_{D}(u,\eta)=a(u,\eta)-\int_{D}\tilde{\eta}\,d\mu,\quad\eta\in H^{\alpha}_{0}(D)$, 2. (b)

$\int_{D}(u-\eta)\,d\mu=0$* for every quasi-continuous function $\eta$ on $D$ such that $u\leq\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e.,* 3. (iii)

There exists a càdlàg process $M$ with $M_{0}=0$ and a positive measure $\mu\in{\cal M}_{0,b}(D)$ such that

1. (a)

$M$* is a uniformly integrable martingale under the measure $P_{x}$ for q.e. $x\in D$, and*

[TABLE] 2. (b)

For any quasi-continuous function $\eta$ on $D$ such that $u\leq\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e.,

[TABLE]

for q.e. $x\in D$.

Proof.

The equivalence of (iii) and (ii) follows from [29, Proposition 3.16]. (i) $\Rightarrow$ (ii). For $\xi\in C_{c}^{\infty}(D)$, we let

[TABLE]

By (5.1), $I(\xi)\geq 0$ for $\xi\geq 0$. Therefore, by Riesz’s theorem, there exists a positive Radon measure $\nu$ on $D$ such that $I(\xi)=\int_{D}\xi\,d\mu$, $\xi\in C_{c}^{\infty}(D)$. Hence

[TABLE]

From this one can easily conclude that $\mu\in S_{0}(D)$ and that the above equation holds for any quasi-continuous $\eta\in H^{\alpha}_{0}(D)$. By (5.1) and (5.2),

[TABLE]

for any quasi-continuous $\eta\in H^{\alpha}_{0}(D)$ such that $\eta\leq\mathbb{I}_{D\setminus D_{0}}$ a.e. Thus, (ii)(b) follows. The implication (ii) $\Rightarrow$ (i) is trivial. ∎

Proposition 5.3**.**

If $u$ is a weak solution to (1.4), then $u\in C_{0}(D)$.

Proof.

Follows from [30, Proposition 4.2]. ∎

5.2. Existence and asymptotics

Let us recall the definition of a weak solution to (1.6) (see Remark 4.2).

Definition 5.4**.**

We say that a strictly positive function $u_{p}\in H^{\alpha}_{0}(D)$ is a weak solution to (1.6) if

[TABLE]

Theorem 5.5**.**

1. (i)

For every $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$ there exists a unique bounded weak solution $u$ to (1.4). 2. (ii)

Let $u_{p}$, $p>1$, be a weak solution to (1.6). Then

[TABLE]

Proof.

Choose $\{q_{k}\}\subset{\cal B}^{+}_{b}(D)$ so that supp$[q_{k}]\subset D\setminus\overline{D}_{0},\,k\geq 1$, and for every compact $K\subset D\setminus\overline{D}_{0}$, $\inf_{x\in K}q_{k}(x)\nearrow\infty$. Let $\psi_{k}$ be the principal eigenfunction for $-(\Delta^{\alpha})_{|D}+q_{k}$. Then for any $c\geq 0,\,k\geq 1$,

[TABLE]

By the fact that $a<\lambda_{1}^{D_{0}}$ and Theorem 3.1, there exists $k_{0}\geq 1$ such that $\lambda_{1}^{D}[q_{k_{0}}]-a\geq 0$. Observe that

[TABLE]

where $K_{k_{0}}=\mbox{supp}[q_{k_{0}}]$. Since $K_{k_{0}}$ is compact and $K_{k_{0}}\subset D\setminus\overline{D}_{0}$, (H1) implies that

[TABLE]

Hence, since $q_{k_{0}}$ is bounded, there exists $c_{k_{0}}$ such that for any $p\geq 2$,

[TABLE]

Therefore $c_{k_{0}}\psi_{k_{0}}$ is a supersolution to (1.6). Since $\lambda_{1}^{D}<a$, we easily conclude that $\varepsilon\varphi^{D}_{1}$ is a subsolution to (1.6) for a sufficiently small $\varepsilon>0$. Moreover, by [16, Theorem 3.4], for a sufficiently small $\varepsilon>0$, $\varepsilon\varphi^{D}_{1}\leq c_{k_{0}}\psi_{k_{0}}$. By Propositions 4.4 and 4.5,

[TABLE]

By the definition of a weak solution to (1.6),

[TABLE]

where $\mu_{p}=bu^{p}_{p}\cdot m$. Taking $\eta=u_{p}$ as a test function and using (5.6), we conclude that

[TABLE]

From this and (5.7) we deduce that $\sup_{p\geq 2}\|\mu_{p}\|_{H^{-\alpha}(D)}<\infty$ (cf. (2.14)). Therefore, there exists $u\in H^{\alpha}_{0}(D)$ and $\mu\in S_{0}(D)$ such that, up to a subsequence, $\mu_{p}\rightarrow\mu$ weakly in $H^{-\alpha}(D)$ and $u_{p}\rightarrow u$ weakly in $H^{\alpha}_{0}(D)$. Moreover, since $H^{\alpha}_{0}(D)$ is compactly embedded in $L^{q}(D;m)$ for $q\in[1,2d(d-2\alpha))$ (cf. Section 2.1), up to a subsequence, $u_{p}\rightarrow u$ a.e. From the second inequality in (5.8) and (H1), we easily deduce that

[TABLE]

Taking $\eta=u_{p}-u$ in (5.7) we get

[TABLE]

Next, by (5.9),

[TABLE]

Substituting into (5.10), and using weak convergence of $\{u_{p}\}$ in $D({\cal E}_{D})$ and strong convergence of $\{u_{p}\}$ in $L^{q}(D;m)$ we conclude that, up to a subsequence,

[TABLE]

Let $\eta\in H^{\alpha}_{0}(D)$ be such that $\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e., and let $\gamma\in(0,1)$. By (H1),

[TABLE]

Therefore, by already proved convergence properties of $\{u_{p}\}$, and (5.7), we get that for every $\eta\in H^{\alpha}_{0}(D)$ such that $\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e.,

[TABLE]

This together with (5.9) implies that $u$ is a weak solution to (1.4). Moreover, by Proposition 5.3, $u$ is continuous. By the uniqueness result for (1.4) (see [30]), ${\cal E}_{D}(u_{p}-u,u_{p}-u)\rightarrow 0$.

As for the uniform convergence in (ii), by Proposition 5.2(iii), and Lemma 2.1,

[TABLE]

for some martingales $M^{p},M$, and q.e. $x\in D$. By Itô’s formula, (H1) and (2.13),

[TABLE]

for any $q\in(1,d/(d-2\alpha))$, and with $c$ depending only on $q,D,\alpha$ and $d$. Set $\theta:=c_{k_{0}}\|\psi_{k_{0}}\|_{\infty}$. By (5.6) and ultracontractivity of $(P^{D}_{t})_{t\geq 0}$ (cf. (2.22)), for $h>0$ we have

[TABLE]

Since $\mu\in S_{0}(D)$ as shown above, we have

[TABLE]

By Proposition 5.2(iii), for every $x\in D$,

[TABLE]

Since $(P^{D}_{t})$ is Fellerian, $\|P^{D}_{h}(u)-u\|_{\infty}\rightarrow 0$ as $h\searrow 0$. Consequently, putting together (5.2)–(5.2), and the already proven convergence properties of $(u_{p})$, we conclude that $\|u_{p}-u\|_{\infty}\rightarrow 0$ as $p\rightarrow\infty$. ∎

6. Parabolic equations: existence and probabilistic interpretation

Let $m_{1}$ be the Lebesgue measure on ${\mathbb{R}}^{d+1}$. Set $D_{T}=(0,T)\times D$. Let $\langle\cdot,\cdot\rangle$ denote the duality pairing between $H^{\alpha}_{0}(D)$ and its dual space $H^{-\alpha}(D)$. Set

[TABLE]

[TABLE]

and define a bilinear form ${\cal B}^{D}$ by

[TABLE]

Let $\mathfrak{X}^{D}=((\mathscr{X}^{D}_{t})_{t\geq 0},(P_{s,x})_{(s,x)\in{\mathbb{R}}\times D},(\mathscr{F}_{t})_{t\geq 0})$ be a Hunt process associated with the form $\mathcal{B}^{D}$ (see [39, Theorem 6.3.1]). In fact (see [39, Theorem 6.3.1] again)

[TABLE]

where $\upsilon(t)$ is the uniform motion to the right, i.e. $\upsilon(t)=\upsilon(0)+t$ and $\upsilon(0)=s$ $P_{s,x}$-a.s. Moreover, $X^{D}$ is a càdlàg process such that for any Borel subset $B$ of $D$,

[TABLE]

It follows that for fixed $s\geq 0$ process $t\mapsto X^{D}_{s+t}$ under measure $P_{s,x}$, for $x\in D$, agrees with the process $\mathbb{X}^{D}$ introduced in Section 2.2.

As in [39, Section 6.2], we define a Choquet capacity naturally associated with the form ${\cal B}^{D}$. We shal denote it by $\mbox{Cap}_{1}$. Then, as in the case of the form ${\cal E}_{D}$, we define quasi-notions associated with $\mbox{\rm Cap}_{1}$ ($\mbox{Cap}_{1}$-q.e., ${\cal B}^{D}$-quasi-continuity, ${\cal B}^{D}$-smooth measures). We denote by ${\cal M}_{0,b}({\mathbb{R}}\times D)$ the set of ${\cal B}^{D}$-smooth bounded measures on ${\mathbb{R}}\times D$, and for fixed $T>0$, we denote by ${\cal M}_{0}(D_{T})$ the subset of ${\cal M}_{0,b}({\mathbb{R}}\times D)$ consisting of measures $\mu$ such that $\mu(({\mathbb{R}}\times D)\setminus D_{T})=0$. By [32, Proposition 4.1], for every positive smooth measure $\mu$ on $D_{T}$ there exists a unique PNAF (positive natural additive functional) $A^{\mu}$ of $\mathfrak{X}^{D}$ in the Revuz duality with $\mu$.

In the sequel, for a function $v$ on $D_{T}$, we let

[TABLE]

and for a given measure $\mu\in{\cal M}_{0,b}(D_{T})$, we denote by $\mu^{(T)}$ the measure on $D_{T}$ given by

[TABLE]

Recall that, starting from Section 3, we assume in the paper that conditions (H1),(H2) (see Introduction) are satisfied. In the sequel, we frequently use, without special mention, that ${\cal W}(0,T)\subset C([0,T];L^{2}(D;m))$ (see e.g. [34, Remarque 1.2, page 156]).

6.1. Probabilistic interpretation of solutions to linear equations

Let $\varphi\in L^{2}(D;m)$ and $f\in L^{2}(D_{T};m_{1})$. Consider the following linear equation.

[TABLE]

Definition 6.1**.**

We say that a bounded function $v\in{\cal W}(0,T)$ is a weak solution to (6.1) on $[0,T]$ if $v(0,\cdot)=\varphi$ and for every $\eta\in L^{2}(0,T;H^{\alpha}_{0}(D))$,

[TABLE]

Proposition 6.2**.**

Let $\varphi\in L^{2}(D;m)$ and $f\in L^{2}(D_{T};m_{1})$.

1. (i)

$v$* is a weak solution to (6.1) if and only if*

[TABLE] 2. (ii)

There exists a unique weak solution $v$ to (6.1) on $[0,T]$. Moreover, $v$ is a strong solution to (6.1), i.e. $v$ is absolutely continuous on $[0,T]$, $\frac{dv}{dt}\in L^{1}(0,T;L^{2}(D;m))$, $v(t)\in D((\Delta^{\alpha})_{|D})$ a.e. $t\in[0,T]$, and

[TABLE] 3. (iii)

Let $\tilde{v}^{(T)}(s,x)$ be equal to the right-hand side of (6.3) if it is finite, and $\tilde{v}^{(T)}(s,x)=0$ otherwise. Then there exists a càdlàg process $M$ with $M_{0}=0$ such that $M$ is an $(\mathscr{F}_{t})_{t\geq 0}$-martingale under the measure $P_{s,x}$ and

[TABLE]

for every $(s,x)\in D_{T}$ such that $\mathbb{E}_{s,x}|\varphi(X^{D}_{T})|+\mathbb{E}_{s,x}\int_{0}^{T-s}|f^{(T)}(\mathscr{X}^{D}_{r})|\,dr<\infty$.

Proof.

(i) and (iii) follow from [27, Theorem 3.7, Theorem 5.8]. Observe that (6.3) means that $v$ is a mild solution to (6.1). Therefore, by [50, Theorem 8.2.1], $v$ is a strong solution to (6.1). ∎

Remark 6.3**.**

For brevity (and in light of Proposition 6.2(ii)), we frequently write that $v$ is a weak (strong) solution to

[TABLE]

instead of writing that it is a weak (strong) solution to (6.1).

Remark 6.4**.**

The displayed formula in Proposition 6.2(iii) says that the pair of processes $(\tilde{v}^{(T)}(\mathscr{X}^{D}),M)$ is a solution of the so called Backward Stochastic Differential Equation (BSDE) with terminal condition $\varphi(X^{D}_{T})$, and right-hand side $f^{(T)}(\mathscr{X}^{D})$ (see [31]).

6.2. Existence for parabolic logistic equations

Theorem 6.5**.**

For every $p>0$ there exists a unique bounded weak solution $v_{p}$ to (1.1). Moreover, $\frac{dv_{p}}{dt},(\Delta^{\alpha})_{|D}v_{p}\in C((0,T];L^{2}(D;m))$.

Proof.

The existence of a bounded weak solution to (1.1) follows from [27, Theorem 3.7,Theorem 5.4,Theorem 5.8]. The uniqueness part is a standard result (see e.g. [27, Proposition 3.6]). By [35], the semigroup $(P^{D}_{t})$ is analytic on $L^{2}(D;m)$. Therefore, by [41, Theorem 3.1, Section 4.3], $[\varepsilon,T]\ni t\mapsto v_{p}(t)\in L^{2}(D;m)$ is $\frac{1}{2}-$Hölder continuous for any $\varepsilon>0$. Now, the asserted regularity follows from [41, Theorem 3.5, Section 4.3]. ∎

Remark 6.6**.**

Let $v_{p}\in{\cal W}(0,T)$ be a bounded function. In light of the above theorem, (6.2) is equivalent to each of the following statements: (a) for any $t\in(0,T)$ and $\eta\in H^{\alpha}_{0}(D)$,

[TABLE]

(b) $\frac{dv_{p}}{dt}\in L^{2}(D_{T})$, $v_{p}(t)\in D((\Delta^{\alpha})_{|D})$ a.e. $t\in(0,T)$, $v_{p}(0)=\varphi$, and

[TABLE]

7. Obstacle problem and asymptotics as $p\rightarrow\infty$ for parabolic equations

Let ${\cal W}_{T}(0,T)=\{u\in{\cal W}(0,T):u(T)=0\}$. In this section, we shall prove asymptotics, with respect to the increasing power of the absorption term, for parabolic logistic equations. To this end, as in the elliptic case, we begin with providing some equivalent formulations of the parabolic obstacle problem (1.2). We shall also show some regularity results for weak solutions to (1.2).

Recall that we assumed that (H1), (H2) are satisfied and $D$ is a bounded Lipschitz domain.

7.1. Obstacle problem

Definition 7.1**.**

We say that $v\in C([0,T];L^{2}(D;m))\cap L^{2}(0,T;H^{\alpha}_{0}(D))$ is a weak solution to (1.2) on $[0,T]$ if

1. (i)

$v\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. and $v(0,\cdot)=\varphi$ a.e., 2. (ii)

For every $\eta\in{\cal W}(0,T)$ such that $\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. we have

[TABLE]

Remark 7.2**.**

It is an elementary check that if additionally to regularity of $v$ required in the definition of weak solution to (1.2), we know that $v\in{\cal W}(0,T)$, then (ii) is equivalent to the following condition: for every $\eta\in L^{2}(0,T;H^{\alpha}_{0}(D))$ such that $\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. we have

[TABLE]

Furthermore, if we know that $\frac{dv}{dt}\in L^{2}(D_{T};m_{1})$, then the above condition is equivalent to the following one: for every $\eta\in H^{\alpha}_{0}(D)$ such that $\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ $m$-a.e. we have

[TABLE]

Before we proceed to the next result, we recall some auxiliary notions. A function $u:D_{T}\to{\mathbb{R}}$ is called *quasi-càdlàg *(see [28]) if for q.e. $(s,x)\in D_{T}$ process $u(\mathscr{X}^{D})$ is càdlàg on $[0,T-s]$ under measure $P_{s,x}$. In [28] (see definition on page 704 in [28] and comments following it), we introduced a notion of a probabilistic solution to (1.2) according to which, $u:D_{T}\to{\mathbb{R}}$ is a solution to (1.2) if $u\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e., there exists a positive smooth measure $\nu$ on $D_{T}$ such that

[TABLE]

and for any quasi-càdlàg function $\eta$ such that $u\leq\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e., we have

[TABLE]

Here $\hat{u},\hat{\eta}$ are precise versions of $u,\eta$, respectively. By the very definition of the precise version (see definition on page 692 in [28]; see also comments preceding Lemma 5.1 in [40]), if $u,\eta$ are quasi-continuous, then $\hat{u}=u$ and $\hat{\eta}=\eta$. In this case (7.2) may be replaced by the following condition: for any quasi-continuous function $\eta$ such that $u\leq\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e., we have

[TABLE]

By [28, Theorem 5.3] (see also the comments after definition on page 704 in [28]), formulation (7.1),(7.3) guarantees the uniqueness of a quasi-continuous probabilistic solution to (1.2).

Proposition 7.3**.**

(1) Assume that $v$ is quasi-continuous and $v\in L^{2}(0,T;H^{\alpha}_{0}(D))$. Then the following statements (i)–(iii) are equivalent.

1. (i)

$v$* is a weak solution to (1.2) on $[0,T]$.* 2. (ii)

There exists a positive $\nu\in{\cal M}_{0,b}(D_{T})$ such that

1. (a)

\int_{0}^{T}\big{\langle}\frac{d\eta}{dt},v\big{\rangle}\,dt+\int_{0}^{T}{\cal E}_{D}(v,\eta)\,dt=(\varphi,\eta(0))+a\int_{0}^{T}(v,\eta)\,dt-\int_{D_{T}}\tilde{\eta}\,d\nu* for every bounded $\eta\in{\cal W}_{T}(0,T)$,* 2. (b)

$\int_{D_{T}}(v-\eta)\,d\nu=0$* for every quasi-continuous function $\eta$ on $D_{T}$ such that $v\leq\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e.* 3. (iii)

There exists a càdlàg process $M$ with $M_{0}=0$ and a positive measure $\nu\in{\cal M}_{0,b}(D_{T})$ such that

1. (a)

$M$* is a uniformly integrable $(\mathscr{F}_{t})_{t\geq 0}$-martingale under the measure $P_{s,x}$ for q.e. $(s,x)\in D_{T}$, and*

[TABLE] 2. (b)

For every quasi-continuous function $\eta$ on $D_{T}$ such that $v^{(T)}\leq\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e.,

[TABLE]

(2) Let $v_{n}\in{\cal W}(0,T)$ be a quasi-continuous version of a weak solution to

[TABLE]

Then, defining $\bar{v}:=\liminf_{n\rightarrow\infty}v_{n}$, we get that $\bar{v}^{(T)}$ satisfies (iii)(a). Moreover, if $\bar{v}$ is quasi-continuous, then $\bar{v}^{(T)}$ satisfies (iii)(b).

Proof.

By the proof of [28, Theorem 5.4] (see Eq. (5.4) in [28]), $\bar{v}^{(T)}(\mathscr{X}^{D})$ satisfies (iii)(a), and (7.2), where $\bar{v}$ is defined in (2). If $\bar{v}$ is quasi-continuous, then by the comment following (7.2), we have that (7.3) holds. Therefore, by the definition of the Revuz duality, we get (iii)(b). This completes the proof of (2).

From [27, Proposition 3.6, Theorem 3.7, Theorem 5.8] it follows that (ii)(a) and (iii)(a) are equivalent, whereas from the definition of the Revuz duality between $\nu$ and $A^{\nu}$ it follows that (ii)(b) and (iii)(b) are equivalent. Therefore (ii) is equivalent to (iii). The proof of (1) shall be completed by showing that (i) is equivalent to (iii). To do this end, we first note that by [34, Theorem 6.2, Chapter 3], there exists a unique weak solution $v$ to (1.2), and it is the limit of functions $v_{n}$ solving (7.4). Suppose that $v$ is a solution to $(iii)$. By [28, Theorem 5.4], $\bar{v}^{(T)}(\mathscr{X}^{D}_{t})=v^{(T)}(\mathscr{X}_{t}),\,t\in[0,T-s],\,P_{s,x}$-a.s. for q.e. $(s,x)\in D_{T}$, where $\bar{v}$ is defined in (2). Thus, $v$ ($=\bar{v}$ a.e.) is a weak solution to (1.2). Suppose now, that $v$ is a weak solution to (1.2). As already mentioned $v=\bar{v}$ a.e. Therefore, for any $(s,x)\in D_{T}$,

[TABLE]

By (2), $\bar{v}^{(T)}$ satisfies (iii)(a), and so $\bar{v}^{(T)}(\mathscr{X}^{D})$ is a càdlàg process. Since $v$ is assumed to be quasi-continuous, we have that $v^{(T)}(\mathscr{X}^{D})$ is a càdlàg process too (see the comments preceding Lemma 5.1 in [40]). Therefore, using the above equation, we conclude that $\hat{v}^{(T)}(\mathscr{X}^{D}_{t})=v^{(T)}(\mathscr{X}^{D}_{t}),\,t\in[0,T-s],\,P_{s,x}$-a.s. for q.e. $(s,x)\in D_{T}$. This implies, in particular, that $\bar{v}=v$ q.e., and so $\bar{v}$ is quasi-continuous. As a result, applying (2), we get that $v^{(T)}(\mathscr{X}^{D})$ satisfies (iii)(b). By [28, Proposition 5.7], $\nu$ is bounded. So, $v$ satisfies (iii). ∎

In the sequel we will freely use, without special mention, the equivalent notions of solutions to (1.2) stated in Proposition 7.3, depending on the source we will refer to.

Remark 7.4**.**

Proposition 7.3(iii) says that the triple $(\tilde{v}^{(T)}(\mathscr{X}^{D}),M,A^{\nu^{(T)}})$ is a solution of the so called Reflected Backward Stochastic Differential Equation (RBSDE) with terminal condition $\xi:=\varphi(X^{D}_{T})$, right-hand side $f(y):=ay$, and barrier $L_{t}:=\mathbb{I}_{D\setminus\overline{D}_{0}}(\mathscr{X}^{D}_{t})$ (see [26]).

In the following proposition, we use the notion of perfect PCAFs of $\mathfrak{X}^{D}$ (see [5, Section IV] for the definition).

Proposition 7.5**.**

Let $v$ be a weak solution to (1.2) on $[0,T]$. Then $v$ is quasi-continuous and there exists a perfect PCAF $\tilde{A}^{\nu^{(T)}}$ of $\mathfrak{X}^{D}$ such that condition (iii) of Proposition 7.3 holds for every $(s,x)\in D_{T}$. Moreover, if $\varphi\in C_{0}(D)$, then $v\in C([0,T];C_{0}(D))$.

Proof.

By [28, Proposition 5.5]), $v\leq w$ , where $w$ is a weak solution of the Cauchy-Dirichlet problem

[TABLE]

It is clear that $w$ is bounded, so $v$ is bounded, too. Set

[TABLE]

By [39, Theorem 6.3.1], $h$ is quasi-continuous. By Proposition 7.3(2),

[TABLE]

Thus, $v^{(T)}\leq h\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. Therefore, in fact, $v$ is a weak solution to (1.2) with $\mathbb{I}_{D\setminus\overline{D}_{0}}$ replaced by $h$. Now, applying Proposition 7.3(iii) (see also Remark 7.4) to $v$ with $\mathbb{I}_{D\setminus\overline{D}_{0}}$ replaced by $h$, and then [26, Corollary 4.4], we conclude that $A^{\nu^{(T)}}$ is continuous $P_{s,x}$-a.s. for q.e. $(s,x)\in D_{T}$. Therefore, by [47, Theorem IV.3.8], $v$ is quasi-continuous. Set

[TABLE]

where

[TABLE]

[TABLE]

By Proposition 7.3(iii) and Revuz duality, for q.e. $(s,x)\in D_{T}$ we have

[TABLE]

Since $\tilde{v}^{(T)}_{2}$ is lower semi-continuous (as $p_{D}$ is lower semi-continuous), the above inequality holds for every $(s,x)\in D_{T}$. Therefore, by [5, Theorem IV.3.13, Theorem V.2.1], there exists a perfect PCAF $\tilde{A}^{\nu^{(T)}}$ such that

[TABLE]

Since $\tilde{v}^{(T)}=v^{(T)}$ a.e., we have

[TABLE]

Applying now a standard argument (see [27, Theorem 5.8]) shows that condition (iii) of Proposition 7.3 holds for every $(s,x)\in D_{T}$ with $A^{\nu^{(T)}}$ replaced by $\tilde{A}^{\nu^{(T)}}$. Assume that $\varphi\in C_{0}(D)$. Set

[TABLE]

Since $h\leq\hat{h}\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$, $v$ is a weak solution to (1.2) with $\mathbb{I}_{D\setminus\overline{D}_{0}}$ replaced by $\hat{h}$. Observe that

[TABLE]

Since $(P^{D_{0}}_{t})$ is strongly Feller (as $\mathbb{X}$ is strongly Feller), $\hat{h}\in C(D)$. By probabilistic interpretations of $\tilde{v}_{n}$ (cf. (7.4)) and $\tilde{v}$ (see Proposition 7.3(2), Proposition 6.2(ii)), and [48, Theorem 1], $v\in C([0,T];C(D))$. Since $v\leq w$, and $w\in C([0,T];C_{0}(D))$, by classical results, $v\in C([0,T];C_{0}(D))$ as well. ∎

7.2. Existence and asymptotics

Theorem 7.6**.**

1. (i)

For every $a\geq 0$ there exists a unique weak solution $v$ to (1.2). 2. (ii)

Let $v_{p}$, $p>0$, be a weak solution to (1.1). Then for every $\delta\in(0,T]$,

[TABLE] 3. (iii)

If, in addition, $\varphi\in H^{\alpha}_{0}(D)$, then $\frac{dv_{p}}{dt}\rightarrow\frac{dv}{dt}$ weakly in $L^{2}(D_{T};m_{1})$, and if $\varphi\in C_{0}(D)$, then (7.5) holds with $\delta=0$.

Proof.

Part (i) follows from [34, Theorem 6.2, Chapter 3]. By [27, Corollary 5.9],

[TABLE]

By Proposition 6.2,

[TABLE]

Therefore, by (7.6) and [17, Lemma 94, page 306], there exists a subsequence (still denoted by $\{v_{p}\}$) such that $\{v_{p}\}$ is convergent a.e. From this and (7.6), we infer that for all $q\geq 1$ and $T\geq 0$, $\{v_{p}\}$ converges in $L^{q}(D_{T})$ to some $v\in L^{q}(D_{T})$. Taking $\eta=v_{p}$ as a test function in (6.2) we obtain

[TABLE]

Hence, up to a subsequence, $v_{p}\rightarrow v$ weakly in $L^{2}(0,T;H^{\alpha}_{0}(D))$. Observe also that, since $\sup_{p\geq 2}\int_{0}^{s}bv_{p}^{p+1}\,dt<\infty$, we have $v\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. Taking $\eta=\frac{dv_{p}}{dt}$ in (6.2) we get

[TABLE]

The rest of the proof we divide into two steps.

Step 1. We assume additionally that $\varphi\in H^{\alpha}_{0}(D)$. Then, by (7.6) and (7.2),

[TABLE]

From this we conclude that $\frac{dv}{dt}\in L^{2}(D_{T};m_{1})$ and, up to a subsequence, $\frac{dv_{p}}{dt}\rightarrow\frac{dv}{dt}$ weakly in $L^{2}(D_{T};m_{1})$. Now, taking $\eta=v_{p}-v$ as a test function in (6.2), we find

[TABLE]

Since $v\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e., we have

[TABLE]

which converges to zero as $p\rightarrow\infty$. Substituting the above inequality into the previous one and using already proven convergences of the sequence $\{v_{p}\}$, we conclude that

[TABLE]

as $p\rightarrow\infty$. From this and a parabolic counterpart of the argument given in (5.2) we infer that $v$ is a weak solution to (1.2). Applying now a uniqueness argument shows the convergence of the whole sequence $\{v_{p}\}$.

To prove the uniform convergence of $\{v_{p}\}$ in (7.5), we first assume additionally that $\varphi\in C_{0}(D)$. Then, since $(P^{D}_{t})_{t\geq 0}$ is Fellerian, a fixed point argument shows that $v_{p}\in C([0,T];C_{0}(D))$. By Proposition 6.2 and Proposition 7.5 (we drop superscript $\tilde{\,}\,\,$),

[TABLE]

for $(s,x)\in D_{T}$. By Itô’s formula

[TABLE]

So, by (H1) and (2.21), there exists $q>1$ such that

[TABLE]

By ultracontractivity of $(P^{D}_{t})_{t\geq 0}$ (cf. (2.22)), for $h>0$, we have

[TABLE]

Taking $\eta=|v_{p}-v|$ as a test function in Proposition 7.3(ii) (remind here that, as shown above, $v_{p},v\in H^{1}(0,T;L^{2}(D;m))$, so $|v_{p}-v|\in H^{1}(0,T;L^{2}(D;m))\subset{\cal W}(0,T)$) and using the already proven convergences of $\{v_{p}\}$ shows that the right-hand side of the above inequality tends to zero as $p\rightarrow\infty$. Next, by (7.6) and Proposition 7.5,

[TABLE]

Observe that

[TABLE]

Since $v\in C([0,T];C_{0}(D))$ (see Proposition 7.5), then using the Feller property of $(P^{D}_{t})$ shows that

[TABLE]

Since we know that $v_{p}\rightarrow v$ in $L^{q}(D_{T})$, from (7.2) and the estimates following it, we deduce that $\|v_{p}-v\|_{\infty}\rightarrow 0$ as $p\rightarrow\infty$.

Step 2. The general case. Let $\varphi_{\varepsilon}\in H^{\alpha}_{0}(D)\cap C_{0}(D)$ be a positive bounded function such that $\|\varphi_{\varepsilon}-\varphi\|_{L_{2}(D;m)}\leq\varepsilon$ and $\varphi_{\varepsilon}\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. Let $v$ be a weak solution to (1.2), $v_{p}^{\varepsilon}$ be a weak solution to (1.1) with $\varphi$ replaced by $\varphi_{\varepsilon}$, and $v^{\varepsilon}$ be a weak solution to (1.2) with $\varphi$ replaced by $\varphi_{\varepsilon}$. By a standard argument,

[TABLE]

[TABLE]

Hence

[TABLE]

It follows from this and Step 1 that the second term in (7.5) tends to zero as $p\rightarrow\infty$. For the uniform convergence in (7.5), let $v_{n}$ be a weak solution of (7.4) and $v^{\varepsilon}_{n}$ be a weak solution of (7.4) with $\varphi$ replaced by $\varphi_{\varepsilon}$. Set $w^{\varepsilon}_{n}=v_{n}-v^{\varepsilon}_{n}$. Observe that

[TABLE]

where $F_{n}(t,y)=-n(y+v^{\varepsilon}_{n}-\mathbb{I}_{D\setminus\overline{D}_{0}})^{+}+n(v^{\varepsilon}_{n}-\mathbb{I}_{D\setminus\overline{D}_{0}})^{+}$. By Proposition 6.2(ii) for q.e. $(s,x)\in D_{T}$ and any $t\in[0,T-s]$,

[TABLE]

From this and [31, Lemma 2.3] (see also Remark 6.4), we deduce that

[TABLE]

By Proposition 7.3, and (2.20), for every $s\in[0,T-\delta]$,

[TABLE]

Analogously, we get the above estimate for $|v_{p}^{\varepsilon}(T-s,x)-v_{p}(T-s,x)|$. From this and Step 1, we get the desired result. ∎

8. Asymptotics as $t\rightarrow\infty$

As in Sections 3-7, we assume that the hypotheses (H1), (H2) are satisfied and $D$ is a bounded Lipschitz domain.

8.1. Cauchy-Dirichlet problem

Lemma 8.1**.**

Let $v_{p}$ be a weak solution to (1.1). Assume that there exists $c>0$ such that $b\geq c$ on $D$. Then

[TABLE]

Proof.

Set $M_{p}:=\max\{\|\varphi\|_{\infty},(a/c)^{1/(p-1)}\}$. By Proposition 6.2 and the Tanaka-Meyer formula (see, e.g., [42, IV.Theorem 70])

[TABLE]

the last inequality being a consequence of the fact that $ay-cy^{p}\leq 0$, $y\geq M_{p}$. ∎

Corollary 8.2**.**

Under the assumptions of Lemma 8.1, the reaction measure $\mu$ for the unique weak solution $v$ to (1.2) is of the form $\mu=g\cdot m_{1}$ with some positive $g\in L^{\infty}(\mathbb{R}^{+}\times D)$.

Proof.

By Lemma 8.1, the term $bv_{p}^{p}$ in (1.1) is bounded uniformly in $p\geq 1$. Therefore, applying Theorem 7.6 gives the result. ∎

Lemma 8.3**.**

Let $v_{p}$ be a weak solution to (1.1) and $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$. Then for every $\delta,t_{0}>0$ there exist $M,c>0$ such that

[TABLE]

and

[TABLE]

Proof.

For $\varepsilon>0$, we set $D_{\varepsilon}=\{x\in{\mathbb{R}}^{d}:\mbox{dist}\{x,D\}<\varepsilon\}$. Let $\{q_{k}\}$ be a sequence of bounded positive functions on $D_{\varepsilon}$ such that supp$[q_{k}]\subset D_{\varepsilon}\setminus\overline{D}_{0}$ and for every compact $K\subset D_{\varepsilon}\setminus\overline{D}_{0}$, $\inf_{x\in K}q_{k}(x)\nearrow\infty$. Let $\psi_{k}$ be the principal eigenfunction for $-(\Delta^{\alpha})_{|D_{\varepsilon}}+q_{k}$. As in the proof of Theorem 5.5 we show that for fixed $\delta>0$ there exist $k_{0}\in\mathbb{N}$ and $c_{k_{0}}>0$ such that $\lambda^{D}_{1}[q_{k}]\geq a,\,k\geq k_{0}$, $c\psi_{k_{0}}\geq\varphi^{D_{\varepsilon}}_{1}$ and $c\psi_{k_{0}}$ is a supersolution to (1.6) on $D_{\varepsilon}$ for $c\geq c_{k_{0}},\,p\geq 1+\delta$. More precisely, there exists a positive bounded function $h$ on $D_{\varepsilon}$ such that

[TABLE]

(see the reasoning following (5.5)). Of course, since $\psi_{k_{0}}$ is independent of $t$, we have

[TABLE]

Let $c$ be chosen so that $c\psi_{k_{0}}\geq 1$ on $D$. By Proposition 6.2 and the Tanaka-Meyer formula, for every $t\in[0,T-s]$,

[TABLE]

Applying Gronwall’s lemma gives

[TABLE]

Observe that

[TABLE]

The last equality follows from the fact that $c\psi_{k_{0}}\geq 1$ on $D$. Consequently,

[TABLE]

Taking $t=0$, we get (8.1). Now, let $w_{p}$ be a weak solution to the Cauchy problem

[TABLE]

By [27, Corollary 5.9], $w_{p}\leq v_{p}$. By Lemma 8.1, $h:=\|b\|_{\infty}w^{p-1}_{p}$ is bounded by a constant independent of $p$. Observe that

[TABLE]

By [27, Corollary 5.9], $w_{p}\geq\bar{w}_{p}$, where $\bar{w}_{p}$ is a weak solution to the problem

[TABLE]

By the Feynman-Kac formula,

[TABLE]

By the ultracontractivity of $(P^{D}_{t})$ (cf. (2.23))

[TABLE]

with some $c_{t_{0}}>0$. Now, observe that for any $c>0$,

[TABLE]

For $c\leq(a-\lambda_{1}^{D})^{p-1}/(\|b\|_{\infty}\|\varphi^{D}_{1}\|_{\infty})$, the last term in braces on the right-hand side of the above equation is less than or equal to zero. Therefore, by [27, Corollary 5.9], for such $c>0$ we have $w_{p}(t)\geq(c\wedge c_{t_{0}})\varphi^{D}_{1}$, $t\geq t_{0}$. This completes the proof since $v_{p}\geq w_{p}$ as shown above. ∎

The following simple lemma appears to be very useful in the proofs of the large time asymptotics of solutions to (1.1), (1.2) (see e.g. [51] for the similar technique).

Lemma 8.4**.**

Let $T,a>0$, and $c\in(0,T)$. Assume that $y:(0,\infty)\to[0,\infty)$, $\int_{0}^{\infty}y(r)\,dr<\infty$, and

[TABLE]

Then

[TABLE]

Moreover, if $\int_{0}^{\infty}y(r)\,dr<\infty$, and (8.3) holds with $T=\infty$, then $\lim_{t\to\infty}y(t)=0$.

Proof.

We first prove the second assertion. By (8.3) and integrability assumption on $y$, we have that $c_{\delta}:=\sup_{t\geq\delta}y(t)<\infty$ for any $\delta>0$. Suppose that $\limsup_{t\to\infty}y(t)=\beta>0$. Set $T:=\frac{\beta}{4ac}$. Then there exists an increasing sequence $\{t_{n}\}$ ($t_{1}\geq\delta$) such that $t_{n+1}-t_{n}\geq 2T$, and $y(t_{n})\geq\beta/2,\,n\geq 1$. Consequently, for any $t\in[t_{n+1}-T,t_{n+1}]$,

[TABLE]

This in turn implies that

[TABLE]

which contradicts to integrability of $y$ over $(0,\infty)$.

As to the second assertion, we let $j:{\mathbb{R}}\to{\mathbb{R}}$ be a positive smooth function with compact support in $B(0,1)$ such that $j(0)=1$, and $\int_{{\mathbb{R}}}j(t)\,dt=1$. Set $j_{\varepsilon}(t):=\frac{1}{\varepsilon}j(t/\varepsilon),\,t\in{\mathbb{R}}$, and $\hat{y}_{\varepsilon}(t):=y_{\varepsilon}(t+\varepsilon)$, with

[TABLE]

Using this notation and (8.3), we find that

[TABLE]

and hence in turn that

[TABLE]

By [51, Lemma 1.1],

[TABLE]

Equivalently,

[TABLE]

Now, letting $\varepsilon\searrow 0$, we easily get the desired result. ∎

Proposition 8.5**.**

Let $v_{p}$ be a weak solution to (1.1) and $u_{p}$ be a weak solution to (1.6). Then

[TABLE]

Proof.

Let $j:{\mathbb{R}}\to{\mathbb{R}}$ be as in the proof of Lemma 8.4. For any $u\in L^{1}_{loc}(D_{T})$ (extended by zero on $D_{T}^{c}$) we denote

[TABLE]

Observe that (cf. Theorem 6.5)

[TABLE]

Multiplying the above equation by $\frac{dv^{(\varepsilon)}_{p}}{dt}$ and integrating over $D$ we find

[TABLE]

Integrating over $[t_{1},t_{2}]\subset[\varepsilon,T-\varepsilon]$, and letting $\varepsilon\searrow 0$ yields

[TABLE]

Write $y_{p}(t)=\|\frac{dv_{p}}{dt}(t)\|^{2}_{L^{2}(D;m)}$. Then

[TABLE]

Since $v_{p}(t)\in H^{\alpha}_{0}(D)$ for $t>0$, we conclude from (7.2) and Lemma 8.3 that $\int_{\varepsilon}^{\infty}y_{p}(t)\,dt<\infty$ for any $\varepsilon>0$. Therefore, by Lemma 8.4, $y_{p}(t)\rightarrow 0$. By (7.2) and Lemma 8.3 again, $\sup_{t\geq t_{0}}{\cal E}_{D}(v_{p}(t),v_{p}(t))<\infty$ for every $t_{0}>0$. As a result, there exists a sequence $\{t_{n}\}\subset{\mathbb{R}}^{+}$ such that $t_{n}\rightarrow\infty$ and $\{v_{p}(t_{n})\}$ is convergent as $n\rightarrow\infty$ to some $w_{p}\in H^{\alpha}_{0}(D)$ weakly in $H^{\alpha}(D)$ and strongly in $L^{q}(D;m)$ for any $q\geq 1$. Since $v_{p}$ is a weak solution to (1.1), we have (cf. Remark 6.6)

[TABLE]

for any $\eta\in H^{\alpha}_{0}(D)$. Letting $n\rightarrow\infty$ we obtain

[TABLE]

for any $\eta\in H^{\alpha}_{0}(D)$. By Lemma 8.3, $w_{p}$ is strictly positive, and so by Proposition 4.5, $w_{p}=u_{p}$. By the uniqueness argument, the convergence of $\{v_{p}(t_{n})\}$ can be strengthened to the convergence of $v_{p}(t)$ as $t\rightarrow\infty$. Now, subtracting (6.5) from (5.4), and then taking $\eta=v_{p}(t)-u_{p}$ as a test function, and using the already proven convergences of $\{v_{p}(t)\}$ as $t\rightarrow\infty$, we deduce that $v_{p}(t)\rightarrow u_{p}$ in $H^{\alpha}(D)$ as $t\rightarrow\infty$.

To prove the uniform convergence in (8.5), we first observe that

[TABLE]

Consequently, by Proposition 6.2 and the Tanaka-Meyer formula, for every $x\in D$,

[TABLE]

Equivalently,

[TABLE]

for every $x\in D$. By (2.20) and (8.1),

[TABLE]

Let $0<S<T$. We have

[TABLE]

By (8.1) and ultracontractivity of $(P^{D}_{t})_{t\geq 0}$ (cf. (2.22)),

[TABLE]

By (2.13), for any $q\in[1,d/(d-2\alpha))$

[TABLE]

Summing up the above inequalities, we conclude that for any $x\in D$,

[TABLE]

Since $v_{p}(t)\rightarrow u_{p}$ in $L^{q^{*}}(D;m)$ as shown above, we conclude from (8.1), by letting $T\to\infty$ and then $S\to\infty$, that $\|v_{p}(T)-u_{p}\|_{\infty}\rightarrow 0$ as $T\rightarrow\infty$. ∎

8.2. Obstacle problem

Theorem 8.6**.**

Let $\varphi\in H^{\alpha}_{0}(D)$, $v$ be a weak solution to (1.2) and $u$ be a weak solution to (1.4). Then

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Proof.

We divide the proof into two steps.

Step 1. Convergence in the energy norm. By (7.2) and (8.1),

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Therefore, under the notation of the prof of Proposition 8.5,

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By (8.6) and Lemma 8.4 for any $\delta>0$,

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Set $k_{p}(t):=-y_{p}(t)+2a\int_{0}^{t}y_{p}(r)\,dr,\,t>0$. By (8.6), $k_{p}$ is non-decreasing, and

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By (8.10), (8.11), there exist $y\in L^{1}(0,\infty)$ and non-increasing right-continuous function $k$ such that

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Set $\hat{y}(t):=2a\int_{0}^{t}y(r)\,dr-k(t),\,t>0$. Clearly, $\hat{y}(t)=\limsup_{p\to\infty}y_{p}(t)=y(t)$ a.e. $t>0$. Thus,

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Hence

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By (8.10), $\hat{y}\in L^{1}(0,\infty)$, and so, by Lemma 8.4,

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By Theorem 7.6(iii),

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Thus,

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By Theorem 7.6 and (8.9), $\mbox{ess\,sup}_{t\geq 0}{\cal E}_{D}(v(t),v(t))<\infty.$ Consequently, there exist a sequence $\{t_{n}\}$, and $u\in H^{\alpha}_{0}(D)$ such that $t_{n}\rightarrow\infty$, $\frac{dv}{dt}(t_{n})\rightarrow 0$ in $L^{2}(D;m)$, $v(t_{n})\rightarrow u$ weakly in $H^{\alpha}_{0}(D)$ and for every $q\geq 1$, $v(t_{n})\rightarrow u$ in $L^{q}(D;m)$. We may assume that $\{t_{n}\}$ is chosen so that (cf. Remark 7.2)

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for every $\eta\in H^{\alpha}_{0}(D)$ such that $\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ a.e. Letting $n\rightarrow\infty$ and using (8.2) shows that $u$ is a weak solution to (1.4). By the uniqueness for (1.4) (see [30]) and the fact that $v\in{\cal W}(0,T)\subset C([0,T];L^{2}(D;m))$, $T\geq 0$, we have $v(t)\rightarrow u$ as $t\rightarrow\infty$ strongly in $L^{2}(D;m)$. By (6.5),

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From (8.11) and Theorem 7.6, we infer that $v\in C((0,T];H^{\alpha}_{0}(D))$, and for all $s,t\geq 1$,

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From this and already proven properties of $v$, we conclude that $v(t)\rightarrow u$ in $H^{\alpha}_{0}(D)$ as $t\rightarrow\infty$.

Step 2. The uniform convergence in (8.8). By Theorem 5.5 and Theorem 7.6, letting $p\rightarrow\infty$ in (8.1) we get

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for $x\in D$. By the asymptotics proved in the first step, we conclude at once from the above inequality that $\|v(T)-u\|_{\infty}\rightarrow 0$ as $T\rightarrow\infty$. ∎

Acknowledgements

This work was supported by Polish National Science Centre (Grant No. 2017/25/B/ST1/00878).