Global Estimates and Regularity of Retarded Parabolic Equations with Fast-growing Nonlinearities

TL;DR

This paper investigates the global estimates and regularity of solutions to retarded parabolic equations with fast-growing nonlinearities, highlighting the relationship between dissipative structures and solution regularity in bounded domains.

Contribution

It provides new insights into the regularity and estimates of solutions for retarded parabolic equations with fast-growing nonlinearities, emphasizing the role of dissipative structures.

Findings

Established global estimates for solutions.

Demonstrated regularity results under dissipative conditions.

Revealed connections between dissipative structures and solution regularity.

Abstract

This paper is concerned with global estimates and regularity of solutions for the initial value problem of the retarded parabolic equation $$\frac{\patial u}{\patial t}-\Delta u=f(x,u)+g(u(x,t-r_1(t)),\cdots,u(x,t-r_m(t)))+h(x,t)$$ in a bounded domain $\Omega\subset R^n$ with fast-growing nonlinearities and a dissipative structure, which is associated with the homogeneous Dirichlet boundary condition. Our results reveal some deeper inherent connections between dissipative structures and the regularity of solutions for such problems.

Equations

$$\frac{\partial u}{\partial t}-\Delta u=f(x,u)+g(u(x,t-r_1(t)),\cdots ,u(x,t-r_m(t)))+h(x,t)$$

$$\begin{array}{rl}\frac{\partial u}{\partial t}-\Delta u=& f(u)+g(u(x,t-r_1(t)),\cdots ,u(x,t-r_m(t)))+h(x,t)\end{array}$$

$$u(x,t)=0,x\in \partial \Omega ,$$

$$f(s)s\le -\Lambda |s{|}^{\gamma +1}+N,\forall s\in \mathbb{R}.$$

$$|f^{\prime }(s)|\le a_0(|s{|}^{\alpha -1}+1),\forall s\in \mathbb{R}.$$

$$|\nabla g(v)|\le b_0(|v{|}^{\beta -1}+1),\forall v=(v_1,\cdots ,v_m)\in {\mathbb{R}}^m.$$

$$u(\cdot ,s)=\varphi (s),s\in [-r,0]$$

$$p_{\ast }=\beta (\gamma -1)/(\gamma -\beta ),q_{\ast }=\max (p_{\ast },2\alpha ,2\beta ).$$

$$u\in C\left([-r,\infty );V_1\right)\bigcap L^{\infty }\left((-r,\infty );V_1\right)\bigcap L^{\infty }\left((-r,\infty );L^q(\Omega )\right)$$

$$\varphi \in C\left([-r,0];V_1\right)\bigcap L^{\infty }\left((-r,0);L^q(\Omega )\right),$$

$$\left\{\begin{array}[]{ll}h\in L^{\infty}(\mathbb{R}\times\Omega)\bigcap L^{\infty}(\mathbb{R};H^{1}(\Omega)),\\[4.30554pt] h^{\prime}\in L^{2}((a,b);H),\hskip 14.22636pt\forall\,-\infty<a<b<\infty,\end{array}\right.$$

$$\left\{\begin{array}[]{ll}u\in C([-r,\infty);V_{2})\bigcap L^{\infty}((-r,\infty);V_{2}),\\[4.30554pt] u^{\prime}\in L^{2}((0,T);V_{1})\bigcap C([0,T];H),\hskip 14.22636pt\forall\,0<T<\infty,\end{array}\right.$$

$$\varphi \in C\left([-r,0];V_2\right)\bigcap L^{\infty }\left((-r,0)\times \Omega \right)$$

$$g(v)=g_1(v_1)+g_2(v_2)+\cdots +g_m(v_m),v\in {\mathbb{R}}^m.$$

$$\varphi \in C\left([-r,0];V_2\right)\bigcap L^{\infty }\left((-r,0);L^q(\Omega )\right),{\int }_{-r}^0{\int }_{\Omega }|\varphi {|}^{q-2}|\nabla \varphi {|}^{2}dxdt<\infty$$

$$\begin{array}{r}|\Delta u{|}^2\le {\tilde{M}}_2\left(\Vert \varphi {\Vert }_{C\left([-r,0];V_2\right)}^2+\Vert \varphi {\Vert }_{L^{\infty }\left((-r,0);L^q(\Omega )\right)}^q+{\int }_{\Omega }|\varphi {|}^{q-2}|\nabla \varphi {|}^{2}dx\right)e^{-{\lambda }_{2}t}+{\tilde{\rho }}_2.\end{array}$$

$$(a_1+a_2+\cdots +a_m{)}^p\le m^p(a_1^p+a_2^p+\cdots +a_m^p).$$

$$ab\le \epsilon a^p+{\epsilon }^{-q/p}{b}^q.$$

$$ab\le \epsilon a^p+\frac{1}{q{p}^{q/p}}{\epsilon }^{-q/p}{b}^q\le \epsilon a^p+{\epsilon }^{-q/p}{b}^q.$$

$${\int }_{\Omega }|u{|}^p|v|dx\le \epsilon |u{|}_q^q+{\epsilon }^{-p/(q-p)}{\left|v\right|}_{(q-p)/p}^{(q-p)/p}.$$

$${\int }_{\Omega }|u{|}^{p}dx\le \epsilon |u{|}_q^q+{\epsilon }^{-p/(q-p)}|\Omega |.$$

$$\begin{array}[]{ll}\lim_{t\rightarrow+\infty}E(t+s,s)=0\mbox{ uniformly w.r.t. $s\in\mathbb{R}^{+}$},\end{array}$$

$$\begin{array}[]{ll}\kappa:=\sup_{t\geq 0}\int_{0}^{t}K(t,s)ds<\infty.\end{array}$$

$$\begin{array}[]{ll}y(t)\leq&E(t,\tau)\|y_{\tau}\|_{\mathcal{C}}+\int_{\tau}^{t}K(t,s)\|y_{s}\|_{\mathcal{C}}ds+\rho,\hskip 28.45274pt\forall\,t\geq\tau\geq 0,\end{array}$$

$$y_t(s)=y(t+s),s\in [-r,0].$$

$${\mathcal{L}}_r(E;K;\rho )=\{y\in C([-r,\infty )):y\ge 0\text{mbox}andsatisfies\text{eqref}e1.3\}.$$

$$y(t)<\mu \rho +\epsilon ,t>T$$

$$\mu =1/(1-\kappa ).$$

$$y(t)\le M\Vert y_0{\Vert }_{\mathcal{C}}{e}^{-\lambda t}+\eta \rho ,t\ge 0$$

$$\eta =(\mu +1)/(1-\kappa c),c=\max \left(\vartheta /(1-\kappa ),1\right).$$

Full text

**Global Estimates and Regularity of Retarded

Parabolic Equations with Fast-growing

Nonlinearities 111This work was supported by the National Natural Science Foundation of China [11871368]. **

Desheng Li

School of Mathematics, Tianjin University

Tianjin 300072, China

*E-mail: lidsmath@tju.edu.cn (D. Li), *

Abstract. This paper is concerned with global estimates and regularity of solutions for the initial value problem of the retarded parabolic equation

[TABLE]

in a bounded domain $\Omega\subset\mathbb{R}^{n}$ with fast-growing nonlinearities and a dissipative structure, which is associated with the homogeneous Dirichlet boundary condition. Our results reveal some deeper inherent connections between dissipative structures and the regularity of solutions for such problems. Keywords: Regularity, dissipativity, retarded parabolic equation 2010 MSC: 35B40, 35B41, 35B65, 35K20, 35K58

1 Introduction

This paper is basically concerned with global estimates and regularity of solutions for the initial value problem of the following retarded parabolic equation:

[TABLE]

with fast growing nonlinearities in a bounded domain $\Omega\subset\mathbb{R}^{n}$, which is associated with the homogeneous Dirichlet boundary condition:

[TABLE]

where $f\in C^{1}(\mathbb{R}),\,g\in C^{1}(\mathbb{R}^{m})$, and $h$ is a measurable function on $\Omega\times\mathbb{R}$. The delay functions $r_{i}$ belongs to $C(\mathbb{R};[0,r])$ ($1\leq i\leq m$) for some $0\leq r<\infty$. This type of problems have been studied by many authors in the past decades; see e.g. [1, 2, 3, 6, 8, 11, 13, 14, 17, 18, 19, 20]. However, due to technical difficulties induced by time lags in the equations, we find that most of the existing works mainly focus on the case where the delay terms have at most sublinear nonlinearities.

In this present work we are interested in the case where both the dominant term $f$ and the delay term $g$ in (1.1) may have arbitrary polynomial growth rates. Since we make no restrictions on the dimension of the space, the investigation of qualitative properties such as the global existence and uniqueness, the regularity, and particularly the dynamics of the equation in functional spaces with higher regularities is in fact never an easy task even if we come back to the situation of the non-delayed case; see e.g. [2, 9, 10, 12, 15, 16] etc. Our main purpose is to carry out a systematic study on the aforementioned problem under a typical dissipative type condition on $f$.

Specifically, let $f$ and $g$ satisfy the following structure conditions:

1. (F0)

There exist $\Lambda,N>0$ and $\gamma>1$ such that

[TABLE] 2. (F1)

There exists ${\alpha}\geq 1$ and $a_{0}>0$ such that

[TABLE] 3. (G1)

There exist $1\leq\beta<\gamma$ and $b_{0}>0$ such that

[TABLE]

For simplicity in writing, we assign the initial time $\tau=0$ for (1.1)-(1.2) and write $u(t)=u(\cdot,t)$ for a solution $u(x,t)$ (in the distribution sense) of the problem with initial value

[TABLE]

Set

[TABLE]

Denote $h(t)=h(\cdot,t)$.

Given $q_{*}<q\leq\infty$, suppose $h\in L^{\infty}(\mathbb{R};L^{q_{\gamma}/\gamma}(\Omega))$, where $q_{\gamma}=q-1+{\gamma}$. We show that (1.1)-(1.3) has a unique global solution

[TABLE]

for any initial value function

[TABLE]

where $V_{1}=H^{1}_{0}(\Omega)$. More importantly, we establish exponential decay estimates in $V_{1}\bigcap L^{q}(\Omega)$ in case $q<\infty$ and uniform boundedness estimates in $V_{1}\bigcap L^{q}(\Omega)$ in case $q=\infty$.

Concerning the regularity properties of $u$ and the dissipativity of the problem in functional spaces with higher regularities, we have the following interesting results: Assume $r_{i}\in C^{1}(\mathbb{R})$ $(1\leq i\leq m)$, and suppose that

[TABLE]

Then

[TABLE]

provided that

[TABLE]

and $\phi^{\prime}\in L^{2}((-r,0);H)$, where $V_{2}=H^{2}(\Omega)\bigcap H^{1}_{0}(\Omega)$. A particular but important case in applications is the one where the time lags $r_{i}$ ($1\leq i\leq m$) are separated, i.e. the function $g$ takes the form

[TABLE]

In such a case we can show that if

[TABLE]

for some $q_{*}<q<\infty$ and $\phi^{\prime}\in L^{2}((-r,0);H)$, then the regularity results in (1.6) remain true. What is more, for each fixed $q>q_{*}$, there exist positive constants $\tilde{M}_{2},\lambda_{2}$ and $\tilde{\rho}_{2}$ such that

[TABLE]

It is interesting to note that for $1\leq q\leq q_{*}$ (and particularly for $1\leq q\leq p_{*}$), the global existence of (1.1)-(1.3) remains an open question. To the best of our knowledge, this is the case even if for the non-delayed parabolic equations with nonlinearities as considered here.

This paper is organized as follows. In Section 2 we do some preliminaries, and in Section 3 we establish global $L^{q}$ and $H^{1}$ estimates. Section 4 is devoted to global $H^{2}$ estimates. In Section 5 we state our main results on the global existence and uniqueness, and the regularity of solutions of the problem.

2 Preliminaries

This section is concerned with some preliminaries. We first recall some fundamental inequalities. Then we give an abstract form of the initial value problem of system (1.1)-(1.2).

2.1 Some fundamental inequalities

First, we have the following easy facts.

Lemma 2.1

Let $p>0$. Then for any $a_{1},a_{2},\cdots,a_{m}\geq 0$,

[TABLE]

Lemma 2.2

Let $p,q>1$, and $1/p+1/q=1$. Then for any $a,b,\varepsilon>0$,

[TABLE]

Proof. This is a simple consequence of the classical Young’s inequality. Indeed,

[TABLE]

This is precisely what we desired. $\Box$

Using the Hölder’s inequality and Lemma 2.2, one trivially verifies the validity of the lemma below.

Lemma 2.3

Let $0<p<q<\infty$, and let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Then for any $u\in L^{q}(\Omega)$, $v\in L^{(q-p)/p}(\Omega)$ and $\varepsilon>0$,

[TABLE]

In particular,

[TABLE]

Here and below $|\Omega|$ denotes the Lebesgue measure of $\Omega$.

The following two retarded integral inequalities given in a recent paper [7] by Li et al. will play a crucial role in our argument.

Let $E$ be a bounded nonnegative measurable function on $Q:=(\mathbb{R}^{+})^{2}$ satisfying

[TABLE]

and let $K$ be a nonnegative measurable function on $Q$ with

[TABLE]

Given $r\geq 0$, denote ${{\mathcal{C}}}$ the space $C([-r,0])$ equipped with the usual sup-norm $\|\cdot\|_{\mathcal{C}}$. Consider the retarded integral inequality

[TABLE]

where $\rho\geq 0$ is a constant, $y$ is a nonnegative continuous function, and $y_{t}$ denotes the element in ${{\mathcal{C}}}$,

[TABLE]

Denote ${\mathscr{L}}_{r}(E;K;\rho)$ the solution set of (2.3), namely,

[TABLE]

Lemma 2.4

[7, Theorem 1.3]**   The the following two assertions hold.

1. $(1)$

If $\kappa<1$ then for any $R,\varepsilon>0$, there exists $T>0$ such that

[TABLE]

for all $y\in{\mathscr{L}}_{r}(E;K;\rho)$ with $\|y_{0}\|_{\mathcal{C}}\leq R$, where

[TABLE] 2. $(2)$

*If $\kappa<1/(1+\vartheta)$, where $\vartheta=\sup_{t\geq s\geq 0}E(t,s)$, then there exist $M,\lambda>0$ *$($independent of \rho$$) such that

[TABLE]

for all $y\in{\mathscr{L}}_{r}(E;K;\rho)$, where

[TABLE]

Remark 2.5

If $\kappa<1/(1+\vartheta)$ then one trivially verifies that $\kappa c<1.$

Remark 2.6

In most examples from applications, the function $E(t,s)$ in (2.3) is an exponential function,

[TABLE]

where $M_{0}$ and $\lambda_{0}$ are positive constants. In such a case we infer from [7, Remark 2.2] that the constants $M$ and $\lambda$ in 2.7 can be taken as

[TABLE]

where $M_{1}=\max\left(\mbox{ln\,}(M_{0}\eta),\,\mbox{ln\,}\left(\frac{2M_{0}}{1-\kappa c}\right)\right).$ In particular, if $r=0$ then we have

[TABLE]

Lemma 2.7

[7, Lemma 2.1]** Suppose $\kappa<1$. Let $y$ be a nonnegative continuous function on $[-r,T)$ $(0<T\leq\infty)$ satisfying

[TABLE]

Then

[TABLE]

where $\mu$ and $c$ are the constants defined in Theorem 3.1.

2.2 Abstract form of problem (1.1)-(1.2)

Let $H=L^{2}(\Omega)$, $V_{1}=H^{1}_{0}(\Omega)$, and $V_{2}=H^{2}(\Omega)\bigcap H^{1}_{0}(\Omega)$. Denote by $(\cdot,\cdot)$ and $|\cdot|$ the inner product and norm on $L^{2}(\Omega)$, respectively, and define the norms $||\cdot||_{1}$ on $V_{1}$ and $||\cdot||_{2}$ on $V_{2}$ as follows:

[TABLE]

By the basic theory on fractional powers of spaces (see e.g. [5, Chap. 1.4]), the $\|\cdot\|_{i}$ is equivalent to the usual norm on $V_{i}$. We will also use $|\cdot|_{q}$ to denote the norm of $L^{q}(\Omega)\ (0<q\leq\infty)$. Denote $A$ the operator $-\Delta$ subjects to the homogeneous Dirichlet boundary condition, and $\mu_{k}$ $({k\geq 1})$ the distinct eigenvalues of $A$,

[TABLE]

Let $X$ be a Banach space $X$. Given $r>0$, denote ${\mathcal{C}}_{X}$ and ${\mathcal{L}}^{\infty}_{X}$ the spaces $C([-r,0],X)$ and $L^{\infty}([-r,0],X)$ equipped with the usual norms:

[TABLE]

respectively. To deal with differential equations with and without delays in a uniform manner, we also assign

[TABLE]

In case $X=L^{q}(\Omega)$ ($0<q\leq\infty$), we will simply write

[TABLE]

The lift of a function $u\in L^{\infty}([\tau-r,T),X)$ ($T>\tau$) in ${\mathcal{L}}^{\infty}_{X}$ is defined to be a mapping $u_{t}$ from $[\tau,T)$ to ${\mathcal{L}}^{\infty}_{X}$,

[TABLE]

Now we define informally a mapping $G(t,\phi)$ in $\mathbb{R}\times{\mathcal{C}}_{V_{1}}$ as below:

[TABLE]

Set $h(t)=h(\cdot,t)$. Then problem (1.1)-(1.2) can be put into an abstract form:

[TABLE]

Since equation (2.11) is nonautonomous, one has to take into account the initial time when considering its initial value problem. Hence the initial value problem of the equation generally reads as

[TABLE]

where $\phi\in{\mathcal{C}}_{V_{1}}$, and $\tau\in\mathbb{R}$ denotes the initial time. Rewriting $t-\tau$ as $t$, one obtains an equivalent form of (2.12):

[TABLE]

where

[TABLE]

Given $\tau\in\mathbb{R}$ and $\phi\in{\mathcal{C}}_{V_{1}}$, denote $u(t;\tau,\phi)$ the solution $u$ of (2.13) (if exists) in the distribution sense on a maximal existence interval $[-r,T_{\phi})$ ($T_{\phi}>0$). For convenience, we call the lift $u_{t}$ of $u$ the solution curve of (2.13) in ${\mathcal{C}}_{V_{1}}$ with initial value $u_{\tau}=\phi$, denoted hereafter by $u_{t}(\tau,\phi)$.

3 $L^{q}$ and $H^{1}$ Decay Estimates

In this section we establish some decay estimates for solutions of the initial value problem (2.13), which in turn imply the regularities of the solutions in appropriate functional spaces.

For simplicity, we only consider the case where the initial time $\tau$ in (2.13) equals [math]. One easily sees that all the estimates given below for $u(t)=u(t;0,\phi)$ hold true for solutions $u(t;\tau,\phi)$ of (2.13) in a uniform manner with respect to $\tau\in\mathbb{R}$.

It should be pointed out that many calculations leading to the estimates are not reasonable because a solution $u$ of (2.13) in the distribution sense may not be sufficiently regular. For instance, in general it remains unknown whether the $L^{q}$-norm $|u(t)|_{q}$ is a continuous function in $t$ for sufficiently large $q$. Hence in the proof of Theorem 3.1 below, Lemmas 3.1 and 2.7 can not be directly applied to $y(t):=|u(t)|_{q}$ to derive decay estimates. However, they can be justified by considering appropriate approximations $u_{k}$ of $u$ as follows.

Let $w_{j}(j=1,2,\cdots)$ be an othorgonal basis of $H=L^{2}(\Omega)$ consisting of eigenvectors of $A$. Given $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{q}$, pick a sequence $\phi_{k}=\sum_{j=1}^{k}a_{j}(t)w_{j}$ and a sequence $h_{k}=\sum_{j=1}^{k}b_{j}(t)w_{j}$ with sufficiently smooth coefficients $a_{j}(t)$ and $b_{j}(t)$ such that $\phi_{k}\rightarrow\phi$ and $h_{k}\rightarrow h$ in appropriate topologies of ${\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{q}$ and $L^{\infty}(\mathbb{R},L^{q}(\Omega))$, respectively. For each $k$, let

[TABLE]

be a Galerkin approximation of (2.13) which solves the following system:

[TABLE]

Here and below $(\cdot,\cdot)$ denotes the inner product of $H=L^{2}(\Omega)$. By the basic theory on ODEs we know that $u_{k}$ is sufficiently regular, so that all the calculations can be performed rigorously on $u_{k}$. As a result, the estimates in the theorems below remain valid for $u_{k}$. Passing to the limit one immediately concludes that these estimates hold true for $u$.

Hence in what follows we always suppose that both the initial value function $\phi$ in (2.13) and the solution $u$ of (2.13) are sufficiently regular when performing mathematical calculations.

3.1 Decay Estimates in $L^{q}(\Omega)$

We first observe that by (F1) and (G1) one has

[TABLE]

and

[TABLE]

Since $\|v\|_{*}=\max_{1\leq i\leq m}|v_{i}|$ is a norm in $\mathbb{R}^{m}$ and all the norms in $\mathbb{R}^{m}$ are equivalent, by (G1) and (3.3) we also have

[TABLE]

and

[TABLE]

Let $p_{*}$ and $q_{*}$ be given as in (1.4), and write

[TABLE]

Denote $u(t;\phi)$ the solution of (2.13) with $\tau=0$ and initial value $\phi$. Our first result is summarized in the following theorem.

Theorem 3.1

Let $q_{*}<q<\infty$. Suppose $h\in L^{\infty}(\mathbb{R};L^{q_{\gamma}/\gamma}(\Omega))$. Then for each $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{q}$, $u(t;\phi)$ is globally defined for $t\geq 0$. Furthermore, there exist $M,\lambda_{q},\rho_{q}>0$, where $M$ is independent of $q$, such that

[TABLE]

Proof. Let $[-r,T_{\phi})$ be the maximal existence interval of $u=u(t;\phi)$. Taking the inner product of both sides of (2.11) with $|u|^{q-2}u$ for $q\geq 2$, we obtain by (F0) and (3.3) that

[TABLE]

Using the Hölder’s inequality and Lemma 2.2 we deduce that

[TABLE]

where $\sigma={q(q-1)}/({q-{\beta}})$, ${\mathcal{C}}_{q}={\mathcal{C}}_{L^{q}(\Omega)}$, and $C_{\varepsilon}=\varepsilon^{-\beta/(q-\beta)}$. One trivially verifies that $\sigma<{q_{\gamma}}$. Thus by Lemma 2.3 we deduce that

[TABLE]

for any $\delta>0$. Taking $\delta=\varepsilon/C_{\varepsilon}$ in the above estimate, it gives

[TABLE]

Therefore

[TABLE]

It is trivial to check that

[TABLE]

We also infer from Lemma 2.3 that

[TABLE]

and

[TABLE]

Combining all the above estimates together we obtain that

[TABLE]

where

[TABLE]

As ${q_{\gamma}}>q$, by Lemma 2.3 one easily deduces that

[TABLE]

We may assume $\Lambda/2\geq\varepsilon{(N+b_{1}^{\prime}(m+1)+1)}$. (3.9) then implies

[TABLE]

where

[TABLE]

In particular,

[TABLE]

Let $E_{\varepsilon}(t,s)=e^{-a_{\varepsilon}(t-s)}$. Clearly for fixed $\varepsilon>0$,

[TABLE]

uniformly w.r.t $s\in\mathbb{R}$. Multiplying (3.13) with $E_{\varepsilon}(t,\tau)$ and integrating in $t$ between $\tau$ and $t$, it gives

[TABLE]

for all $t\geq\tau\geq 0$, where $y(t)=|u(t)|_{q}^{q}$, and $K_{\varepsilon}(t,s)=\varepsilon qmb_{1}^{\prime}E_{\varepsilon}(t,s)$.

Note that $\vartheta:=\sup_{t\geq s}E_{\varepsilon}(t,s)=1$. Since

[TABLE]

we have

[TABLE]

Now we take

[TABLE]

Then $a_{\varepsilon_{0}}>q\Lambda/2$ and

[TABLE]

Thus by Lemma 2.7 we deduce that $|u|_{q}$ is bounded on $[-r,T_{\phi})$. Further using the same argument as in the proof of Theorem 3.5 in Section 3.2 below with minor modifications, it can be shown that $|\nabla u|$ is bounded on $[-r,T_{\phi})$. It then follows that $T_{\phi}=\infty$.

Clearly $C_{\varepsilon_{0}}^{\prime\prime}/a_{\varepsilon_{0}}\leq 2C_{\varepsilon_{0}}^{\prime\prime}/q\Lambda:=\rho_{q}.$ Thus by (3.14) we have

[TABLE]

for $t\geq\tau\geq 0$. Now let us apply Lemma 3.1 to the above inequality. The constants corresponding to those in Lemma 3.1 and Remark 2.6 read

[TABLE]

and

[TABLE]

By virtue of Lemma 3.1 we conclude that

[TABLE]

This completes the proof of the validity of (3.6). $\Box$

Theorem 3.2

Suppose $h\in L^{\infty}(\mathbb{R}\times\Omega).$ Then there exist $\rho_{*},\lambda_{*}>0$ such that

[TABLE]

*for all $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$. *

Proof. Let us first evaluate $\limsup_{q\rightarrow\infty}\rho_{q}^{1/q}$. Note that $q\Lambda/2\leq a_{\varepsilon_{0}}\leq q\Lambda$. Therefore $\lim_{q\rightarrow\infty}a_{\varepsilon_{0}}^{1/q}=1.$ Hence

[TABLE]

On the other hand, by (3.10), (3.12), Lemma 2.1 and (3.8), we deduce that

[TABLE]

where

[TABLE]

Thus we conclude that $\limsup_{q\rightarrow\infty}\rho_{q}^{1/q}\leq\rho_{*}$.

It is trivial to check that if $r>0$ then $\limsup_{q\rightarrow\infty}\lambda_{q}/q=0$; and if $r=0$, by the choice of $\varepsilon_{0}$ we have

[TABLE]

for all $q$ sufficiently large. Therefore using Lemma 2.1 once again, we arrive by (3.17) at the following estimates:

[TABLE]

This is precisely what we desired. $\Box$

Remark 3.3

In case $r=0$ (i.e. for the equation without delay), we infer from Theorem 3.2 that

[TABLE]

for all $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$. However, this remains an open problem in case $r>0$, which may indicate some inherent differences between delay differential equations and those without delays. Fortunately, the following eventual invariance property still remains true.

Proposition 3.4

Assume the hypotheses in Theorem 3.2, and let $\rho_{*}$ be the constant given therein. Then for any $\varepsilon>0$, there is $t_{0}>0$ such that

[TABLE]

for all $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$ with $\|\phi\|_{{\mathcal{L}}^{\infty}_{\infty}}\leq\rho_{*}+\varepsilon$.

Proof. Given $\varepsilon>0$, to prove (3.19), it suffices to check that for any $\rho<\rho_{*}+\varepsilon:=\rho_{*}^{\prime}$, the estimate holds true for each $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$ with $\|\phi\|_{{\mathcal{L}}^{\infty}_{\infty}}\leq\rho$.

First, it is easy to see that

[TABLE]

Hence there is $q_{0}>0$ such that $\lambda_{q}\geq c_{r}$ for all $q>q_{0}$. Take a $t_{0}>0$ such that $Me^{-c_{r}t_{0}}<7/2$. Then

[TABLE]

Since $\limsup_{q\rightarrow\infty}\rho_{q}^{1/q}\leq\rho_{*}$, there exists $q_{1}>q_{0}$ such that

[TABLE]

Let $\|\phi\|_{{\mathcal{L}}^{\infty}_{\infty}}\leq\rho$. We observe that

[TABLE]

Because $\rho<\rho_{*}^{\prime}$ and $|\Omega|^{1/q}\rightarrow 1$ as $q\rightarrow\infty$, we can pick a $q_{2}>q_{1}$ such that $\rho|\Omega|^{1/q}<\rho_{*}^{\prime}$ for all $q>q_{2}.$ It then follows that

[TABLE]

Combing the above estimate together, it yields

[TABLE]

Therefore by (3.22) and (3.24) we deduce that

[TABLE]

The proof of the lemma is complete. $\Box$

3.2 $H^{1}$ decay estimates

In this subsection we give a decay estimate in $V_{1}=H^{1}_{0}(\Omega)$.

Theorem 3.5

Let $q_{*}<q<\infty$. Suppose $h\in L^{\infty}(\mathbb{R},L^{q_{\gamma}/\gamma}(\Omega))$. Then there exist $M_{1},\lambda_{1},\rho_{1}>0$ such that for all $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{q}$,

[TABLE]

Proof. Let $u=u(t;\phi)$. Taking the inner product of (2.11) in $H$ with $-\Delta u$, we obtain that

[TABLE]

By (3.2), (3.5) and the Hölder’s inequality, we deduce that

[TABLE]

and

[TABLE]

Combing these estimates with (3.26) we find that

[TABLE]

Hence by Theorem 3.1 one concludes that there exist $M^{\prime},\lambda^{\prime}>0$ such that

[TABLE]

where $\mu_{1}$ is the first eigenvalue of $A=-\Delta$.

We may assume $\lambda^{\prime}\leq\mu_{1}/2$. Then by the classical Gronwall lemma, there exists $\rho_{1}>0$ such that

[TABLE]

This verifies (4.1). $\Box$

Remark 3.6

Note that for $t\in[-r,0]$, since $u(t)=\phi(t)$ and $e^{-\mu_{1}t}\geq 1$, we see that (3.29) readily holds true. Therefore we actually have

[TABLE]

Remark 3.7

Integrating (3.28) between $t$ and $t+1$, by (3.30) one obtains that

[TABLE]

Similarly, if we integrate (3.11) with $\varepsilon=\varepsilon_{0}$ ($\varepsilon_{0}$ is given by (3.15)) between $t$ and $t+1$, by (3.6) we get

[TABLE]

For $T>0$, integrating (3.11) with $\varepsilon=\varepsilon_{0}$ between [math] and $T$, it also yields

[TABLE]

for some $C_{T}>0$.

4 $H^{2}$ estimates

This section is devoted to the $H^{2}$ decay estimates and global estimates of solutions of (2.13). As in Section3, we may assume the initial time $\tau=0$. This is just for the sake of simplicity in writing, and all the estimates remain true for (2.13) uniformly with respect to $\tau\in\mathbb{R}$.

4.1 The case $\phi\in{\mathcal{L}}^{\infty}_{\infty}$

We first give an $H^{2}$ estimate in case the initial value $\phi\in{\mathcal{L}}^{\infty}_{\infty}$.

Proposition 4.1

Suppose $h\in L^{\infty}(\mathbb{R}\times\Omega)\bigcap L^{\infty}(\mathbb{R};H^{1}(\Omega))$. Then for any $R>0$, there exist $C=C(R)>0$ such that

[TABLE]

for all $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$ with

[TABLE]

Proof. Let $u=u(t;\phi)$. Multiplying (2.11) with $-\Delta u^{\prime}$ (where   ${}^{\prime}=\frac{\partial}{\partial t}$) and integrating over $\Omega$, we get

[TABLE]

where $I_{3}=\int_{\Omega}\nabla h\cdot\nabla u^{\prime}dx$, and

[TABLE]

[TABLE]

Here and below $\partial_{i}g(v)=\frac{\partial}{\partial v_{i}}g(v)$ ($v\in\mathbb{R}^{m}$).

Let us first evaluate $I_{1}$. Fix a $q_{*}<q<\infty$. Using the structure conditions (F1) and (G1), the Hölder’s inequality and the Young’s inequality we deduce that

[TABLE]

Here we have used the simple fact $|u|^{2(\alpha-1)}<|u|^{q-2}+1$. Now assume that $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$ and satisfies (4.2). Then by Theorem 3.2 there exists $\rho_{*}>0$ (independent of $R$) such that

[TABLE]

Hence

[TABLE]

where $e_{1}(t)$ is the function defined in (3.30).

Note that

[TABLE]

By (3.4) we deduce that

[TABLE]

Thus a similar argument as above applies to show that

[TABLE]

We also have

[TABLE]

Therefore by (4.3) one concludes that

[TABLE]

Since $e_{i}(t)$ ($i=1,2,3$) are nonincreasing, we have

[TABLE]

[TABLE]

Applying the Uniform Gronwall Lemma (see Temam [12, pp. 89, Lemma 1.1]) to (4.8) yields

[TABLE]

Taking $t_{0}=r$ one immediately obtains (4.1). $\Box$

Theorem 4.2

Suppose $h\in L^{\infty}(\mathbb{R}\times\Omega)\bigcap L^{\infty}(\mathbb{R};H^{1}(\Omega))$. Then for any $R>0$, there exist $C=C(R)>0$ such that

[TABLE]

for all $\phi\in{\mathcal{C}}_{V_{2}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$ with $\|\phi\|_{{\mathcal{C}}_{V_{2}}},\,\|\phi\|_{{\mathcal{L}}_{\infty}^{\infty}}\leq R.$

Proof. By (4.8) one easily deduces that there is a constant $C=C(R)>0$ such that $|\Delta u(t;\phi)|\leq C$ for $t\in[0,r+1]$. (4.9) then directly follows from this local estimate and Proposition 4.1. $\Box$

4.2 The case of separated delays

In this part we consider a slightly particular case where the delays $r_{i}$ ($1\leq i\leq m$) are separated, namely, $g$ takes the form

[TABLE]

In such a case we establish some $H^{2}$ decay estimates by assuming $\phi\in{\mathcal{L}}^{\infty}_{q}$ for some $q>q_{*}$ rather than $\phi\in{\mathcal{L}}^{\infty}_{\infty}$.

We begin with (4.3). By (4.4) and (3.30) we have

[TABLE]

Using the Hölder’s inequality, the Chauchy-Schwartz inequality, the structure condition (G1) and (3.30), we deduce that

[TABLE]

where $C_{11}=2mb_{0}^{2}$. Combining (4.3), (4.7) and the above two estimates it yields

[TABLE]

By Remark 3.7 we find that

[TABLE]

[TABLE]

Thanks to the Uniform Gronwall Lemma, it follows from (4.11) and the definition of $e_{1}(t),e_{2}(t)$ and $e_{3}(t)$ (see (3.30) and Remark 3.7) that

[TABLE]

for some constants $M_{2}^{\prime},\lambda_{2},\rho_{2}>0$. In particular,

[TABLE]

where $M_{2}=M_{2}^{\prime}e^{\lambda_{2}(r+1)}$. Rewriting $t_{0}+1=t$ in (4.12), we get

Proposition 4.3

Let $q_{*}<q<\infty$. Suppose $h\in L^{\infty}(\mathbb{R}\times\Omega)\bigcap L^{\infty}(\mathbb{R};H^{1}(\Omega))$. Then there exist $M_{2},\lambda_{2}>0$ and $\rho_{2}>0$ such that for all $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{q}$,

[TABLE]

In addition to the hypotheses in Proposition 4.3, if we assume

[TABLE]

then for $T=r+1$, by (3.33) and the definition of $e_{1}(t)$ it is easy to deduce that

[TABLE]

Integrating (4.11) between [math] and $T$ one finds that

[TABLE]

for $t\in[0,T]$. Combining this with Proposition 4.3 it yields the following result.

Theorem 4.4

Assume $g$ takes the form in (4.10). Let $q_{*}<q<\infty$. Suppose $h\in L^{\infty}(\mathbb{R}\times\Omega)\bigcap L^{\infty}(\mathbb{R};H^{1}(\Omega))$. Then there exist positive constants $\tilde{M}_{2}$ and $\tilde{\rho}_{2}$ such that

[TABLE]

for all $\phi\in{\mathcal{C}}_{V_{2}}\bigcap{\mathcal{L}}^{\infty}_{q}$ satisfying (4.14).

5 Existence, Uniqueness and Regularity of Solutions

Using the estimates established in the previous sections, it can be shown by very standard argument via Galerkin approximation methods as stated in the beginning of Section 3 that the following existence and uniqueness result hold true for the initial value problem (2.13).

Theorem 5.1

Let $q_{*}<q\leq\infty$. Suppose $h\in L^{\infty}(\mathbb{R};L^{q_{\gamma}/\gamma}(\Omega))$. Then for each $\phi\in{\mathcal{C}}_{V_{1}}\bigcap{\mathcal{L}}^{\infty}_{q}$, problem (2.13) has a unique global weak solution $u=u(t;\tau,\phi)$ (in the distribution sense) with

[TABLE]

Furthermore, for any $0<T<\infty$,

[TABLE]

Remark 5.2

Since $u\in L^{\infty}\left([-r,\infty);L^{q}(\Omega)\right)$ and $q>q_{*}$, by (3.2) and (3.3) one trivially verifies that

[TABLE]

belongs to $L^{2}([0,T];H)$ for any $T>0$. The relation $u\in C\left([-r,\infty);V_{1}\right)$ in (5.1) then follows from Theorem 3.3 in [12, Chapt. II] on abstract linear equations.

Now we pay some more attention to regularity of solutions when the initial value $\phi$ has higher regularity. Let $w$ be the function in (5.3). For simplicity, as before we set $\tau=0$, hence $h^{\tau}(t)=h(t)$, and

[TABLE]

Therefore

[TABLE]

As above, one easily verify that all the functions $f^{\prime}(u)$ and $\partial_{i}g(u(t-r_{1}),\cdots,u(t-r_{m}))$ ($1\leq i\leq m$) belong to $L^{2}([0,T];H)$ for any $T>0$.

Multiplying (2.11) with $u^{\prime}$ and integrating over $\Omega$, we get

[TABLE]

Using the above inequality and the $H^{1}$ estimates it is not difficulty to see that $u^{\prime}\in L^{2}((0,T);H)$ for any $T>0$. Further if we assume that $\phi^{\prime}\in L^{2}((0,T);H)$ then since $u\in C([-r,T];H)$, it can be shown that $u^{\prime}\in L^{2}((-r,T);H)$.

Now assume $r_{i}\in C^{1}(\mathbb{R})$. Then by (5.4) we deduce that $w^{\prime}\in L^{2}((0,T);H)$. Thanks to Theorem 3.2 in [12, Chapt. II] on regularity of abstract linear equations, and the $H^{2}$ estimates given in Section 4, we obtain the following theorems.

Theorem 5.3

Assume $r_{i}\in C^{1}(\mathbb{R})$ $(1\leq i\leq m)$, and that

[TABLE]

[TABLE]

Then for any $\phi\in{\mathcal{C}}_{V_{2}}\bigcap{\mathcal{L}}^{\infty}_{\infty}$ with $\phi^{\prime}\in L^{2}((-r,0);H)$, the solution $u$ of (2.13) given by Theorem 5.1 satisfies

[TABLE]

[TABLE]

Theorem 5.4

Assume $g$ takes the form in (4.10), and that $r_{i}\in C^{1}(\mathbb{R})$ $(1\leq i\leq m)$. Suppose

[TABLE]

[TABLE]

Then for any $\phi\in{\mathcal{C}}_{V_{2}}\bigcap{\mathcal{L}}^{\infty}_{q}$ $(q_{*}<q<\infty)$ satisfying (4.14) and $\phi^{\prime}\in L^{2}((-r,0);H)$, the solution $u$ of (2.13) given by Theorem 5.1 satisfies

[TABLE]

[TABLE]