Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

Abstract

We study the homogeneous Dirichlet problem for the class of nonlinear parabolic equations with variable nonlinearity $\begin{matrix} u_{t} - div (D (x) | \nabla u |^{p (x) - 2} \nabla u) = f (x, t, u) - A (x) | u |^{q (x) - 2} u \end{matrix}$ in the cylinder $Ω \times (0, T)$ with given nonnegative weights $D (x)$ , $A (x)$ , measurable bounded exponents $p (x) \in [p^{-}, p^{+}]$ , $q (x) \in [q^{-}, q^{+}]$ and a globally Lipschitz function $f (x, t, u)$ . Sufficient conditions of existence and uniqueness of weak and strong solutions are derived. We find conditions on the exponents $p (x)$ , $q (x)$ which guarantee that the associated semigroup has a compact global attractor in $L^{2} (Ω)$ . It is shown that in case the exponents $p (x)$ and $q (x)$ do not meet the sufficient conditions of existence of a nontrivial global attractor and $‖ u (0) ‖_{L^{2} (Ω)}$ is sufficiently small, then every solution with bounded $‖ u (t) ‖_{L^{2} (Ω)}^{2}$ either vanishes in a finite time, or decays exponentially as $t \to \infty$ .

Keywords

Nonlinear parabolic equation variable nonlinearity global attractors

1. Introduction

1.1. Statement of the problem

Let $Ω \subset R^{n}$ be a bounded domain with Lipschitz-continuous boundary $\partial Ω$ and $Q_{T} = Ω \times (0, T)$ , $T > 0$ . We consider the homogeneous Dirichlet problem $\begin{matrix} (1.1) & \begin{matrix} u_{t} - div (D (x) | \nabla u |^{p (x) - 2} \nabla u) + A (x) | u |^{q (x) - 2} u = f (x, t, u) in Q_{T}, \\ u = 0 on \partial Ω \times (0, T), u (x, 0) = u_{0} (x) in Ω . \end{matrix} \end{matrix}$ Throughout the text we assume that $p (x)$ , $q (x)$ are given measurable functions satisfying the conditions $\begin{matrix} (1.2) & 1 < p^{-} ⩽ p (x) ⩽ p^{+} < \infty, 1 < q^{-} ⩽ q (x) ⩽ q^{+} < \infty a.e. in Ω \end{matrix}$ with some constants $p^{\pm}$ , $q^{\pm}$ . About the coefficients $D (x)$ , $A (x)$ and the function f we assume the following:

$D (x)$ , $A (x) \in L^{\infty} (Ω)$ , $D (x)$ , $A (x)$ are nonnegative in Ω;

there is a constant $s > 1$ such that $\begin{matrix} s < p^{-} and D^{\frac{- s}{p (x) - s}} (x) \in L^{1} (Ω); \end{matrix}$

for each $T > 0$ , $f (x, t, s) : Q_{T} \times R \mapsto R$ is globally Lipschitz-continuous with respect to s and has the linear growth: there is a constant L such that $\begin{matrix} (1.3) & \begin{matrix} | f (x, t, u) - f (x, t, v) | ⩽ L | u - v | \forall u, v \in R, (x, t) \in Q_{T}, \\ | f (x, t, s) | ⩽ L | s | + f_{0} (x, t) with some f_{0} \in L^{2} (0, T; L^{2} (Ω)) . \end{matrix} \end{matrix}$

The paper addresses the question of asymptotic behavior of solutions as $t \to \infty$ . We derive conditions on the structure of the equation sufficient for the existence of a compact invariant global attractor and show that in case these conditions are not satisfied, every solution bounded in $L^{2} (Ω)$ either vanishes in a finite time, or decays exponentially as $t \to \infty$ .

1.2. Previous work

The study of existence and the properties of attractors for parabolic problems with variable nonlinearity is a very recent research issue. To the best of our knowledge, the first results were published in [17] where it was shown that the model problem for the evolution $p (x)$ -Laplacian with $D \equiv 1$ , $A \equiv 0$ and the globally Lipschitz forcing term $f \equiv f (u)$ has a global attractor in the range of the exponent $p (x) \in [2 + δ, 3 - δ]$ with a small positive δ. Further results on the equations with globally Lipschitz f can be found in [18,20], see also [15,21] for the study of equations with non-globally Lipschitz f. Global attractors for a coupled system of evolution differential inclusions which involve $p (x)$ -Laplacian operators were considered in [19]. We refer also to [16] for the existence of global attractors for the entropy solutions of parabolic equations with $p (x)$ -Laplacian and $L^{1}$ data and to [11,12] for the results on the pullback attractors for non-autonomous parabolic problems with variable nonlinearity.

The asymptotic behavior of solutions of the Cauchy problem for equation (1.1) with nonnegative weights $A (x)$ , $D (x)$ and the exponents $p (x) \equiv q (x)$ was studied in [2,3], the existence of weak solutions of the Cauchy problem with $p (x) \neq q (x)$ and $p^{-} > 2$ , $q^{-} > 2$ was proved in [2].

In the present work, we do not assume that $q (x)$ coincides with $p (x)$ . In the result, the perturbed weighted $p (x)$ -Laplacian operator is not of subdifferential type, which gives rise to additional difficulties in the proof of the existence of a global attractor. Moreover, since we allow $q (x) < 2$ , this is a singular perturbation. To the best of our knowledge, thus far there are no results on the existence of global attractors for an evolution equation which has a singular perturbation of the $p (x)$ -Laplacian.

The sufficient conditions of existence of a global attractor for the solutions of problem (1.1) are derived by means of the analysis based on the infinite-dimensional dynamical system approach. In the range of the exponents $p (x)$ , $q (x)$ where this approach ceases to be applicable, the study of the asymptotic behavior is performed with the energy method [5,6], which requires a modification due to the special form of equation (1.1).

1.3. Organization of the paper and results

In Section 2 we introduce the weighted Lebesgue and Sobolev spaces with variable exponents and derive the properties of these spaces needed for the study of problem (1.1). These results are of auxiliary character and provide the analytic framework for the further proceeding.

In Section 3 we present conditions of existence and uniqueness of solutions for problem (1.1). In Theorem 3.1 we recall the conditions of existence of weak (energy) solutions. The proof of this assertion is given in paper [2] where problem (1.1) served as an auxiliary problem in construction of a solution of the same equation in $R^{n}$ . In Theorem 3.2 we show that the solutions of problem (1.1) possess better regularity in the class of more regular initial data. Finally, we prove that the weak solution of problem (1.1) is unique.

The main results are given in Sections 4 and 5. In Section 4 we derive sufficient conditions for the existence of a global attractor for problem (1.1). It is shown in Theorem 4.2 that problem (1.1) has a global compact invariant attractor in $L^{2} (Ω)$ if the data satisfy conditions (H1)–(H2) and, additionally, one of the following conditions is fulfilled: $\begin{array}{rcl} (1.4) & \begin{matrix} (a) & p^{-} > s > 2, q^{-} > 1, \\ (b) & p^{-} > s > \frac{2 n}{n + 2} and \{\begin{matrix} either q^{-} > 2, A^{- \frac{2}{q (x) - 2}} (x) \in L^{1} (Ω), \\ or q^{-} > 1, A^{\frac{- 1}{q (x) - 1}} (x) \in L^{1} (Ω), \frac{2}{q^{-}} + \frac{\frac{1}{p^{-}} - \frac{1}{q^{-}}}{1 + \frac{1}{n} - \frac{1}{s}} < 1 . \end{matrix} \end{matrix} \end{array}$

In Section 5 we study the asymptotic behavior of solutions of problem (1.1) with $f_{0} \equiv 0$ in (1.3) and under the assumption that the exponents $p (x)$ , $q (x)$ do not meet conditions (1.4). To be precise, we study the equations with the variable exponents $p (x)$ , $q (x)$ satisfying one of the conditions $\begin{array}{rcl} (1.5) & \begin{matrix} \frac{2 n}{n + 2} < s < p^{-} ⩽ p^{+} < 2, q^{-} > 1, \\ \frac{2 n}{n + 2} < s < p^{-}, q^{-} > 1, A^{\frac{- 1}{q (x) - 1}} (x) \in L^{1} (Ω), \frac{1}{σ^{+}} = \frac{2}{q^{+}} + \frac{\frac{1}{p^{+}} - \frac{1}{q^{+}}}{1 + \frac{1}{n} - \frac{1}{s}} > 1 . \end{matrix} \end{array}$ It turns out that under conditions (1.5) every solution $u (x, t)$ with $‖ u (t) ‖_{2, Ω}^{2} : = ‖ u (t) ‖_{L^{2} (Ω)}^{2}$ bounded uniformly in t must vanish in a finite time, provided that $‖ u_{0} ‖_{2, Ω}$ is sufficiently small (Theorems 5.1, 5.2). Moreover, we show that if conditions (1.5) are fulfilled but a global estimate on $‖ u (t) ‖_{2, Ω}^{2}$ is unavailable, the same property still holds although at expense of more restrictive assumptions on $‖ u_{0} ‖_{2, Ω}$ (Theorems 5.4, 5.5). In the borderline case when in conditions (1.5) $p^{+} = 2$ or $σ^{+} = 1$ , the solutions decay exponentially as $t \to \infty$ .

Conditions (1.4) and (1.5) complement each other. If conditions (1.4) are fulfilled, there exists a compact global attractor. Conversely, if conditions (1.5) hold, then the solutions corresponding to small initial data extinct in a finite time and, thus, the attracting set in $L^{2} (Ω)$ consists of the single element $u = 0$ . For constant p and q conditions (1.4), (1.5) cover the whole range of exponents where the existence of solutions is proven. This fact confirms the necessity of conditions (1.4) for the existence of a nontrivial global attractor of problem (1.1).

All results are established for the nonnegative weights D and A which are not assumed to be strictly positive. The weight functions are subject to the integrability conditions which indicate the admissible order of their zeros. Notice also that unlike most of works on the parabolic PDEs with variable nonlinearity we make no assumptions about the regularity of the exponents $p (x)$ and $q (x)$ .

A prototype of equation (1.1) is furnished by the following example: $\begin{matrix} u_{t} - div (| x |^{β} | \nabla u |^{p (x) - 2} \nabla u) + | x |^{γ} | u |^{q (x) - 2} u = a (t) u, β, γ are nonnegative constants, \end{matrix}$ in a bounded domain Ω that contains the point $x = 0$ . For this equation, condition (1.4) (a) holds with $β < \frac{p^{-}}{s} - 1$ , condition (1.4) (b) with $q^{-} > 1$ is fulfilled if $β < \frac{p^{-}}{s} - 1$ and $γ < n (q^{-} - 1)$ .

Throughout the text we denote by C a generic constant whose exact value may change from line to line and is unimportant (but can be calculated or estimated through the data). We use the notation $z = (x, t)$ for the points of the cylinder $Q_{T} = Ω \times (0, T)$ and denote $Q_{t} = Ω \times (0, t)$ for $t \in (0, T]$ . Whenever it does not cause a confusion, we omit the arguments of the exponents of nonlinearity $p (x)$ , $q (x)$ and the weights $D (x)$ , $A (x)$ .

2. Weighted spaces with variable exponents

2.1. Lebesgue and Sobolev spaces with variable exponents

We begin with a brief description of the standard Lebesgue spaces with variable exponents. A detailed insight into the theory of these spaces and a review of the bibliography can be found in [7,9,13]. Let $Ω \subset R^{n}$ be a bounded domain with the Lipschitz continuous boundary $\partial Ω$ . Define the set $\begin{matrix} P (Ω) = {measurable functions on Ω with values in (1, \infty)} . \end{matrix}$ Given a function $r (x) \in P (Ω)$ , we introduce the functional (the modular) $\begin{matrix} ρ_{r (\cdot), Ω} (f) = \int_{Ω} {| f (x) |}^{r (x)} d x \end{matrix}$ and the set $\begin{matrix} L^{r (\cdot)} (Ω) = {f : Ω \mapsto R : f is measurable on Ω, ρ_{r (\cdot), Ω} (f) < \infty} . \end{matrix}$ The set $L^{r (\cdot)} (Ω)$ equipped with the Luxemburg norm $\begin{matrix} ‖ f ‖_{r (\cdot), Ω} : = inf {α > 0 : ρ_{r (\cdot), Ω} (\frac{f}{α}) ⩽ 1} \end{matrix}$ becomes a Banach space. If $r (x) \in P (Ω)$ and $\begin{matrix} 1 < r^{-} ⩽ r (x) ⩽ r^{+} < \infty a.e. in Ω, \end{matrix}$ then:

$L^{r (\cdot)} (Ω)$ is a reflexive and separable Banach space,

$C_{0}^{\infty} (Ω)$ is dense in $L^{r (\cdot)} (Ω)$ ,

for every $f \in L^{r (\cdot)} (Ω)$ and $g \in L^{r^{'} (\cdot)} (Ω)$ ( $r^{'} = \frac{r}{r - 1}$ is the conjugate exponent of r), the generalized Hölder inequality holds $\begin{matrix} (2.1) & \int_{Ω} | f g | d x ⩽ (\frac{1}{r^{-}} + \frac{1}{{(r^{'})}^{-}}) ‖ f ‖_{r (\cdot), Ω} ‖ g ‖_{r^{'} (\cdot), Ω}, \end{matrix}$

for every $f \in L^{r (\cdot)} (Ω)$ $\begin{matrix} (2.2) & min {ρ_{r (\cdot), Ω}^{\frac{1}{r^{+}}} (f), ρ_{r (\cdot), Ω}^{\frac{1}{r^{-}}} (f)} ⩽ ‖ f ‖_{r (\cdot), Ω} ⩽ max {ρ_{r (\cdot)}^{\frac{1}{r^{+}}} (f), ρ_{r (\cdot), Ω}^{\frac{1}{r^{-}}} (f)}, \end{matrix}$

for every sequence ${f_{k}} \subset L^{r (\cdot)} (Ω)$ and $f \in L^{r (\cdot)} (Ω)$ $\begin{matrix} (2.3) & ‖ f_{k} - f ‖_{r (\cdot), Ω} \to 0 \Leftrightarrow ρ_{r (\cdot), Ω} (f_{k} - f) \to 0 as k \to \infty, \end{matrix}$

since the domain Ω is bounded, $p_{1} (x), p_{2} (x) \in P (Ω)$ and $p_{1} (x) ⩾ p_{2} (x)$ a.e. in Ω, there is a continuous inclusion $L^{p_{1} (\cdot)} (Ω) \subset L^{p_{2} (\cdot)} (Ω)$ and $\begin{matrix} (2.4) & \forall u \in L^{p_{1} (\cdot)} (Ω) ‖ u ‖_{p_{2} (\cdot), Ω} ⩽ C ‖ u ‖_{p_{1} (\cdot), Ω} \end{matrix}$ with a constant $C = C (| Ω |, p_{1}^{\pm}, p_{2}^{\pm})$ .

Let $p (x) \in P (Ω)$ . The variable Sobolev space $W_{0}^{1, p (\cdot)} (Ω)$ is defined as the closure of $C_{0}^{\infty} (Ω)$ with respect to the norm $\begin{matrix} ‖ f ‖_{W_{0}^{1, p (\cdot)} (Ω)} = ‖ \nabla f ‖_{p (\cdot), Ω} + ‖ f ‖_{p (\cdot), Ω} . \end{matrix}$ By (2.4) $W_{0}^{1, p (\cdot)} (Ω) \subset W_{0}^{1, p^{-}} (Ω)$ . If $p^{-} > \frac{2 n}{n + 2}$ , then the embedding $W_{0}^{1, p^{-}} (Ω) \subset L^{2} (Ω)$ is compact and an equivalent norm of $W_{0}^{1, p (\cdot)} (Ω)$ is defined by $\begin{matrix} ‖ f ‖_{W_{0}^{1, p (\cdot)} (Ω)} = ‖ \nabla f ‖_{p (\cdot), Ω} . \end{matrix}$

2.2. Weighted spaces

Let $B (\cdot)$ and $r (\cdot) \in P (Ω)$ be a generic weight function and an exponent. Assume that $\begin{matrix} (2.5) & B \in L^{\infty} (Ω), B ⩾ 0 a.e. in Ω, B^{\frac{- 1}{r (\cdot) - 1}} (x) \in L^{1} (Ω) . \end{matrix}$ Let us define the linear space $\begin{matrix} L_{B (\cdot)}^{r (\cdot)} (Ω) = {f : Ω \mapsto R : f is measurable on Ω, ρ_{r (\cdot), Ω} (B^{\frac{1}{r (\cdot)}} f) < \infty} \end{matrix}$ and the functional $\begin{matrix} ‖ f ‖_{B (\cdot), r (\cdot), Ω} = {‖ B^{\frac{1}{r}} f ‖}_{r (\cdot), Ω} : = inf {λ > 0 : ρ_{r (\cdot), Ω} (\frac{B^{\frac{1}{r (\cdot)}} f (\cdot)}{λ}) < 1} . \end{matrix}$ Under condition (2.5) this functional defines the norm of the space $L_{B (\cdot)}^{r (\cdot)} (Ω)$ .

Proposition 2.1 (The dual space of $L_{B}^{r} (Ω)$ ).

If B satisfies condition ( 2.5 ), then ${(L_{B}^{r} (Ω))}^{'} = L_{b}^{r^{'}} (Ω)$ where $r^{'} = \frac{r}{r - 1}$ , $b = B^{- \frac{1}{r - 1}}$ and the duality is given by $\begin{matrix} ⟨ f, g ⟩ = \int_{Ω} f g d x, f \in L_{B}^{r} (Ω), g \in L_{b}^{r^{'}} (Ω) . \end{matrix}$

Proof.
Since $f \in L_{B}^{r} (Ω) \Leftrightarrow B^{\frac{1}{r}} f \in L^{r} (Ω)$ and ${(L^{r} (Ω))}^{'} = L^{r^{'}} (Ω)$ , every functional $Φ \in {(L_{B}^{r} (Ω))}^{'}$ is identified with a functional $Ψ \in {(L^{r} (Ω))}^{'}$ by means of the relations $\begin{matrix} ⟨ Φ, f ⟩ = ⟨ Ψ, (B^{\frac{1}{r}} f) ⟩ = \int_{Ω} h (B^{\frac{1}{r}} f) d x = \int_{Ω} (B^{\frac{1}{r}} h) f d x \end{matrix}$ with some $h \in L^{r^{'}} (Ω)$ . Since $h \in L^{r^{'}} (Ω) \Leftrightarrow g = B^{\frac{1}{r}} h \in L_{b}^{r^{'}} (Ω)$ , we have the inclusion ${(L_{B}^{r} (Ω))}^{'} \subseteq L_{b}^{r^{'}} (Ω)$ . On the other hand, for every $g \in L_{b}^{r^{'}} (Ω)$ and $f \in L_{B}^{r} (Ω)$ $\begin{array}{l} | \int_{Ω} f g d x | & = | \int_{Ω} (B^{\frac{1}{r}} f) (B^{- \frac{1}{r}} g) d x | ⩽ C {‖ B^{\frac{1}{r}} f ‖}_{r, Ω} {‖ B^{- \frac{1}{r}} g ‖}_{r^{'}, Ω} . \end{array}$ By virtue of (2.2) $‖ f ‖_{B, r, Ω} ⩽ 1 \Leftrightarrow ‖ B^{\frac{1}{r}} f ‖_{r, Ω} ⩽ 1$ , whence $\begin{array}{l} ‖ g ‖_{{(L_{B}^{r} (Ω))}^{'}} & = sup {| ⟨ f, g ⟩ | : ‖ f ‖_{r, B, Ω} = 1} ⩽ C {‖ B^{- \frac{1}{r}} g ‖}_{r^{'}, Ω} \\ ⩽ C max {{(\int_{Ω} b | g |^{r^{'}} d x)}^{\frac{1}{{(r^{'})}^{+}}}, {(\int_{Ω} b | g |^{r^{'}} d x)}^{\frac{1}{{(r^{'})}^{-}}}} . \end{array}$ It follows that $L_{b}^{r^{'}} (Ω) \subseteq {(L_{B}^{r} (Ω))}^{'}$ . □

Let us introduce the weighted Sobolev space W as the closure of $C_{0}^{\infty} (Ω)$ with respect to the norm $\begin{matrix} ‖ f ‖_{W} = {‖ D^{\frac{1}{p}} | \nabla f | ‖}_{p (\cdot), Ω} + ‖ f ‖_{2, Ω} . \end{matrix}$ The dual space $W^{}$ is the set of linear functionals over W equipped with the norm $\begin{matrix} ‖ Φ ‖_{W^{}} = sup {| ⟨ Φ, f ⟩ | : ‖ f ‖_{W} = 1} . \end{matrix}$

Notice that if a weight $D (x)$ satisfies the integrability condition (H2), it satisfies also the inclusion $D^{\frac{- 1}{p (x) - 1}} (x) \in L^{1} (Ω)$ because $\frac{s}{p (x) - s} > \frac{1}{p (x) - 1}$ a.e. in Ω.
Proposition 2.2 (Representation of the functionals in $W^{*}$ ).

A functional Φ belongs to $W^{*}$ if and only if there exist $g_{0} \in L^{2} (Ω)$ and $G \in L_{d}^{p^{'}} (Ω; R^{n})$ with the weight $d (x) = D^{\frac{- 1}{p (x) - 1}} (x)$ such that $\begin{matrix} (2.6) & ⟨ Φ, f ⟩ = \int_{Ω} (g_{0} f + G \cdot \nabla f) d x \forall f \in W . \end{matrix}$

The proof is an adaptation of the proof given in [1] for the usual Sobolev spaces, see also [8] for a study of weighted Sobolev spaces with constant exponents and [10] for the case of Sobolev spaces with variable exponent $p (x, t)$ .

Let us assume that $σ (x) \in C_{\log} (\overline{Ω})$ , that is, $σ (x)$ is continuous in $\overline{Ω}$ with the logarithmic module of continuity: for all $x, y \in Ω$ , $| x - y | < 1 / 2$ , $\begin{matrix} (2.7) & | σ (x) - σ (y) | ⩽ ω (| x - y |), \end{matrix}$ where $ω (s)$ satisfies the condition $\begin{matrix} lim_{s \to 0^{+}} ω (s) ln \frac{1}{s} = τ < \infty . \end{matrix}$

Proposition 2.3 (Theorem 8.3.1, [9]).

Let $r (x), σ (x) \in P (Ω) \cap C_{\log} (\overline{Ω})$ . For every $f \in W_{0}^{1, p (\cdot)} (Ω)$ $\begin{matrix} ‖ f ‖_{r (\cdot), Ω} ⩽ C ‖ \nabla f ‖_{σ (\cdot), Ω} with r (x) < σ^{*} (x) = \{\begin{matrix} \frac{n σ (x)}{n - σ (x)} & if σ (x) < n, \\ any number from [1, \infty) & if σ (x) ⩾ n \end{matrix} \end{matrix}$ and an independent of f constant C.

Let us denote $\begin{matrix} (2.8) & s^{*} = \{\begin{matrix} \frac{n s}{n - s} & for n > s, \\ any number from (1, \infty) & for n ⩽ s \end{matrix} \end{matrix}$ with the constant s from condition (H2).

Proposition 2.4.
Let $r (x) \in P (Ω) \cap C_{\log} (\overline{Ω})$ and $r (x) < s^{}$ in Ω. For every* $f \in W$ $\begin{matrix} (2.9) & ‖ f ‖_{r (\cdot), Ω} ⩽ C {(1 + \int_{Ω} D | \nabla f |^{p} d x)}^{\frac{1}{s}} \end{matrix}$ with an independent of f constant C.
Proof.
By Proposition 2.3 $\begin{matrix} ‖ f ‖_{r (\cdot), Ω}^{s} ⩽ C ‖ \nabla f ‖_{s, Ω}^{s} = C \int_{Ω} | \nabla u |^{s} d x \end{matrix}$ and by Young’s inequality $\begin{array}{l} \int_{Ω} | \nabla f |^{s} d x & = \int_{Ω} {(D | \nabla f |^{p})}^{\frac{s}{p}} D^{- \frac{s}{p}} d x \\ (2.10) & ⩽ \int_{Ω} D | \nabla f |^{p} d x + \int_{Ω} D^{- \frac{s}{p - s}} d x . \end{array}$ □
Remark 2.1.
In the special case $D \equiv const > 0$ we may take $s = p^{-}$ : by the Young inequality $\begin{array}{l} \int_{Ω} | \nabla f |^{s} d x & = \int_{Ω \cap {p (x) = p^{-}}} | \nabla f |^{p^{-}} d x + \int_{Ω \cap {p (x) > p^{-}}} | \nabla f |^{p^{-}} d x \\ ⩽ \int_{Ω \cap {p (x) = p^{-}}} | \nabla f |^{p} d x + \int_{Ω \cap {p (x) > p^{-}}} | \nabla f |^{p} d x + | Ω \cap {p (x) > p^{-}} | \\ ⩽ \int_{Ω} | \nabla f |^{p} d x + | Ω | . \end{array}$

2.3. Spaces of functions depending on x and t

Assume that $p (x) \in P (Ω)$ and let $D (x)$ satisfy conditions (H1)–(H2). Set $Q_{T} = Ω \times (0, T)$ . The Banach space $L_{D}^{p} (Q_{T})$ is the set of functions $\begin{matrix} {u (x, t) : D (x) {| u (x, t) |}^{p (x)} \in L^{1} (Q_{T})} \end{matrix}$ endowed with the norm $\begin{matrix} ‖ u ‖_{D (\cdot), p (\cdot), Q_{T}} = inf {λ > 0 : ρ_{p (\cdot), Q_{T}} (\frac{D^{\frac{1}{p (x)}} (x) u (x, t)}{λ}) < 1} . \end{matrix}$ The space $W (Q_{T})$ is defined as the closure of $C^{\infty} ([0, T]; C_{0}^{\infty} (Ω))$ with respect to the norm $\begin{matrix} ‖ u ‖_{W (Q_{T})} = ‖ \nabla u ‖_{D (\cdot), p (\cdot), Q_{T}} + ‖ u ‖_{2, Q_{T}} . \end{matrix}$ $W^{*} (Q_{T})$ is the topological dual to $W (Q_{T})$ composed of the linear functionals on $W (Q_{T})$ : for every $ϕ \in W^{*} (Q_{T})$ there exist $g_{0}$ and $G = (G_{1}, \dots, G_{n})$ such that $g_{0} \in L^{2} (Q_{T})$ , $D^{- \frac{1}{p}} G_{i} \in L^{p^{'}} (Q_{T})$ and $\begin{matrix} \forall u \in W (Q_{T}) ⟨ u, ϕ ⟩ = \int_{Q_{T}} (g_{0} u + G \cdot \nabla u) d z, z = (x, t) . \end{matrix}$ The norm of $W^{*} (Q_{T})$ is given by $\begin{matrix} ‖ ϕ ‖_{W^{*} (Q_{T})} = sup {⟨ ϕ, u ⟩ : ‖ u ‖_{W (Q_{T})} = 1} . \end{matrix}$

Proposition 2.5.
The dual space $W^{} (Q_{T})$ to $W (Q_{T})$ is isomorphic to the subspace of* $D^{'} (Q_{T})$ composed of the distributions of the form $\begin{matrix} T (ϕ) = \int_{Q_{T}} (g_{0} ϕ + G \cdot \nabla ϕ) d z \end{matrix}$ with $g_{0} \in L^{2} (Q_{T})$ , $G = (G_{1}, \dots, G_{n}) \in L_{d}^{p^{'}} (Q_{T})$ and the weight $d (x) = D^{- \frac{1}{p (x) - 1}} (x)$ .

2.4. Steklov’s means and formulas of integration by parts

For a function $f \in L_{D (\cdot)}^{p (\cdot)} (Q_{T})$ with $p \in P (Ω)$ and the weight $D (x)$ satisfying (H1)–(H2), we define the Steklov mean by $\begin{matrix} f_{h} = \frac{1}{h} \int_{t}^{t + h} f (x, τ) d τ . \end{matrix}$

Proposition 2.6 (Lemmas 2.5, 2.6, 2.7, [2]).

The operator $S_{h} : f \to f_{h}$ maps $L_{D (\cdot)}^{p (\cdot)} (Q_{T})$ into $L_{D (\cdot)}^{p (\cdot)} (Q_{T - h})$ .

The operator of translation in t, $Δ_{h} f (x, t) : = f (x, t + h) - f (x, t)$ , is continuous: $\begin{matrix} {‖ Δ_{h} f (x, t) ‖}_{p (\cdot), D (\cdot), Q_{T - h}} \to 0 as h \to 0 . \end{matrix}$

If $f \in W (Q_{T})$ , then $‖ f_{h} - f ‖_{W (Q_{T - h})} \to 0$ as $h \to 0$ .

The following properties can be derived in the standard way (see [4,10], [5, Chapter 1]) with the use of Propositions 2.5, 2.6:

if $u \in W (Q_{T})$ and $u_{t} \in W^{*} (Q_{T})$ , then ${(u_{t})}_{h} = {(u_{h})}_{t}$ and ${(u_{h})}_{t} ⇀ u_{t}$ in $W^{*} (Q_{T - h})$ as $h \to 0$ ;

if $v, w \in W (Q_{T})$ and $v_{t}, w_{t} \in W^{*} (Q_{T})$ , then for a.e. $t_{1}, t_{2} \in (0, T)$ $\begin{matrix} \int_{t_{1}}^{t_{2}} \int_{Ω} w_{t} v d z + \int_{t_{1}}^{t_{2}} \int_{Ω} w v_{t} d z = \int_{Ω} v w d x |_{t = t_{1}}^{t = t_{2}}; \end{matrix}$

if $u \in W (Q_{T})$ and $u_{t} \in W^{*} (Q_{T})$ , then $u \in C ([0, T]; L^{2} (Ω))$ after possible redefining on a set of zero measure in $(0, T)$ .

3. Existence and uniqueness of weak solutions

Definition 3.1.
A function $u : Q_{T} \mapsto R$ is called weak solution of problem (1.1) if
$u \in C (0, T; L^{2} (Ω)) \cap W (Q_{T})$ , $A (x) | u |^{q (x)} \in L^{1} (Q_{T})$ , $u_{t} \in W^{} (Q_{T})$ ;

for every $ϕ \in C^{1} ([0, T]; C_{0}^{1} (Ω))$ $\begin{matrix} (3.1) & \int_{Q_{T}} (u_{t} ϕ + D (x) | \nabla u |^{p (x) - 2} \nabla u \cdot \nabla ϕ + A (x) | u |^{q (x) - 2} u ϕ - f (x, t, u) ϕ) d z = 0; \end{matrix}$

for every $ϕ \in C_{0}^{1} (Ω) \int_{Ω} ϕ (x) (u - u_{0}) d x \to 0$ as $t \to 0$ ;

the weak solution is called strong solution if $\begin{matrix} u_{t} \in L^{2} (Q_{T}), D | \nabla u |^{p} \in L^{\infty} (0, T; L^{1} (Ω)), A | u |^{q} \in L^{\infty} (0, T; L^{1} (Ω)) . \end{matrix}$

Theorem 3.1.
Let the exponents* $p (x)$ , $q (x)$ satisfy conditions ( 1.2 ), $p^{-} > \frac{2 n}{n + 2}$ and $q^{-} > 1$ . Assume that the coefficients A, D satisfy (H1)–(H2). If $u_{0} \in L^{2} (Ω)$ and f satisfies (H3), then problem ( 1.1 ) has at least one weak solution in the sense of Definition 3.1 . The weak solution satisfies the energy identity: for every $t \in [0, T]$ $\begin{matrix} (3.2) & \frac{1}{2} {‖ u (t) ‖}_{2, Ω}^{2} + \int_{Q_{t}} (D | \nabla u |^{p} + A | u |^{q}) d z = \frac{1}{2} ‖ u_{0} ‖_{2, Ω}^{2} + \int_{Q_{t}} f (x, t, u) u d z . \end{matrix}$ The weak solution satisfies the energy estimate $\begin{matrix} ess sup_{(0, T)} {‖ u (t) ‖}_{2, Ω}^{2} + \int_{Q_{T}} (D | \nabla u |^{p} + A | u |^{q}) d z ⩽ C (‖ u_{0} ‖_{2, Ω}^{2} + ‖ f_{0} ‖_{2, Q_{T}}^{2}) \end{matrix}$ with a constant C which does not depend on u.

Theorem 3.1 is proved in [2, Theorem 5.2]. There are two differences between the present case and the one considered in [2]. First, the proof in [2] is given for the case when Ω is a ball of finite radius and now $Ω \subset R^{n}$ is a Lipschitz domain, which does not alter the arguments. Second, now we claim that $q^{-} > 1$ instead of $q^{-} > 2$ . A revision of the proof in [2] shows that for the bounded domain Ω this difference is unimportant. Theorem 3.2.
Let in the conditions of Theorem 3.1 $u_{0} \in V = W \cap L_{A}^{q} (Ω)$ . Then the weak solution of problem ( 1.1 ) is a strong solution and satisfies the inequality $\begin{matrix} (3.3) & ess sup_{(0, T)} {‖ u (t) ‖}_{2, Ω}^{2} + ess sup_{(0, T)} \int_{Ω} (\frac{D}{p} | \nabla u |^{p} + \frac{A}{q} | u |^{q}) d x + ‖ u_{t} ‖_{2, Q_{T}}^{2} ⩽ C \end{matrix}$ with a constant C depending only on $‖ u_{0} ‖_{V}$ , $‖ f_{0} ‖_{2, Q_{T}}$ and the constants in assumptions (H1)–(H3).
Proof.
The weak solution of problem (1.1) is constructed in [2] as the limit of a sequence of Galerkin’s approximations. The main ingredients of the proof are the uniform a priori estimates for the approximations, monotonicity of the diffusion part of the equation and the formula of integration by parts in t.

To construct the sequence of approximations we take the sequence ${ψ_{i}}$ of the eigenfunctions of the problem $\begin{matrix} {(ψ, ϕ)}_{W_{0}^{l, 2} (Ω)} = λ {(ϕ, ψ)}_{2, Ω} \forall ϕ \in W_{0}^{l, 2} (Ω) \end{matrix}$ with $l ⩾ 1 + n max {0, \frac{1}{2} - \frac{1}{p}}$ , so that $W_{0}^{l, 2} (Ω) ↪ W_{0}^{1, p} (Ω)$ . The set ${ψ_{i}}$ is orthogonal in $W_{0}^{l, 2} (Ω)$ and forms an orthonormal basis of $L^{2} (Ω)$ . A weak solution of problem (1.1) is obtained as the limit of the sequence $u_{m} = \sum_{i = 1}^{m} c_{i, m} (t) ψ_{i} (x)$ , $m \in N$ . The coefficients $c_{i, m} (t)$ are defined from the system of ordinary differential equations $\begin{array}{rcl} (3.4) & \begin{matrix} c_{i, m}^{'} (t) = - {(D (x) | \nabla u_{m} |^{p (x) - 2} \nabla u_{m}, \nabla ψ_{i})}_{2, Ω} \\ - {(A (x) | u_{m} |^{q (x) - 2} u_{m}, ψ_{i})}_{2, Ω} + (f (x, t, u_{m}), ψ_{i}), \\ c_{i, m} (0) = d_{i, m}, i = 1, \dots, m . \end{matrix} \end{array}$ If $d_{i, m} = {(u_{0}, ψ_{i})}_{2, Ω}$ , then $\sum_{i = 1}^{m} d_{i, m} ψ_{i} (x) \to u_{0}$ in $L^{2} (Ω)$ as $m \to \infty$ and the sequence ${u_{m}}$ converges to a weak solution of problem (1.1). Proposition 3.1.
If $u_{0} \in V$ , then there exists a sequence $\begin{matrix} u_{0 m} (x) = \sum_{i = 1}^{m} d_{i, m} ψ_{i} (x) \to u_{0} in V as m \to \infty, d_{i, m} \in R . \end{matrix}$
Proof.
Let us fix an arbitrary $ϵ > 0$ . Since $C_{0}^{\infty} (Ω)$ is dense in V, there exists $w_{ϵ} \in C_{0}^{\infty} (Ω)$ such that $\begin{matrix} (3.5) & \begin{matrix} ‖ u_{0} - w_{ϵ} ‖_{V} & = ‖ u_{0} - w_{ϵ} ‖_{2, Ω} + {‖ \nabla (u_{0} - w_{ϵ}) ‖}_{D, p, Ω} + ‖ u_{0} - w_{ϵ} ‖_{A, q, Ω} < \frac{ϵ}{2} . \end{matrix} \end{matrix}$ The system of eigenfunctions ${ψ_{i}}$ is dense in $W_{0}^{l, 2} (Ω)$ and for every $ϵ > 0$ and $w_{ϵ} (x)$ there is a number $k = k (ϵ)$ and a set of coefficients ${d_{i, k}}$ such that $\begin{matrix} (3.6) & \begin{matrix} {‖ w_{ϵ} - w_{ϵ}^{(k)} ‖}_{V} & ⩽ (1 + ‖ A ‖_{\infty, Ω} + ‖ D ‖_{\infty, Ω}) ({‖ w_{ϵ} - w_{ϵ}^{(k)} ‖}_{q, Ω} + {‖ w_{ϵ} - w_{ϵ}^{(k)} ‖}_{p, Ω}) \\ ⩽ C {‖ w_{ϵ} - w_{ϵ}^{(k)} ‖}_{W_{0}^{l, 2} (Ω)} < \frac{ϵ}{2}, w_{ϵ}^{(k)} = \sum_{i = 1}^{k} d_{i, k} ψ_{i} . \end{matrix} \end{matrix}$ By (3.5), (3.6) and the continuity of the embedding $W_{0}^{l, 2} (Ω) \subset W_{0}^{1, p} (Ω)$ $\begin{matrix} {‖ u_{0} - w_{ϵ}^{(k)} ‖}_{V} & ⩽ ‖ u_{0} - w_{ϵ} ‖_{V} + {‖ w_{ϵ} - w_{ϵ}^{(k)} ‖}_{V} ⩽ \frac{ϵ}{2} + C {‖ w_{ϵ} - w_{ϵ}^{(k)} ‖}_{W_{0}^{l, 2} (Ω)} < \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ . \end{matrix}$ It follows that there is a sequence $u_{0 m} = \sum_{i = 1}^{m} d_{i, m} ψ_{i} (x)$ that converges to $u_{0}$ in the norm of V. □

Let u be a weak solution obtained as the limit of the sequence ${u_{m}}$ . Multiplying each of equations (3.4) by $c_{i}^{'} (t)$ and taking the sum in $i = 1, 2, \dots, m$ we obtain the inequality $\begin{matrix} (3.7) & \begin{matrix} {‖ {(u_{m})}_{t} ‖}_{2, Q_{t}}^{2} & + \int_{Q_{t}} (\frac{D}{p} {(| \nabla u_{m} |^{p})}_{t} + \frac{A}{q} {(| u_{m} |^{q})}_{t}) d x d τ ⩽ \int_{Q_{t}} | f (x, τ, u_{m}) | | {(u_{m})}_{t} | d x d τ . \end{matrix} \end{matrix}$ Let us denote $\begin{matrix} E (u_{m} (t)) = \int_{Ω} (\frac{D}{p} {| \nabla u_{m} (t) |}^{p} + \frac{A}{q} {| u_{m} (t) |}^{q}) d x . \end{matrix}$ Applying the Cauchy inequality to the right-hand side of (3.7) and integrating in t in the second term on the left-hand side we arrive at the estimate $\begin{matrix} \frac{1}{2} {‖ {(u_{m})}_{t} ‖}_{2, Q_{t}}^{2} & + E (u_{m} (t)) ⩽ \frac{1}{2} ‖ f_{0} ‖_{2, Q_{t}}^{2} + E (u_{0 m}) + \frac{L^{2}}{2} {‖ u_{m} (t) ‖}_{2, Q_{t}}^{2} . \end{matrix}$ The norms $‖ u_{m} ‖_{2, Q_{T}}^{2}$ are uniformly bounded – see [2, Lemma 5.2]. According to (2.3) and due to the choice of the sequence ${u_{0 m}}$ , for all sufficiently large m $\begin{matrix} E (u_{0 m}) ⩽ E (u_{0 m} - u_{0}) + E (u_{0}) ⩽ 1 + E (u_{0}) ⩽ C . \end{matrix}$ The final estimate reads $\begin{matrix} {‖ {(u_{m})}_{t} ‖}_{2, Q_{T}}^{2} + ess sup_{(0, T)} E (u_{m} (t)) ⩽ C \end{matrix}$ with a constant C independent of m. Letting $m \to \infty$ and applying the Fatou Lemma we conclude that the weak solution of problem (1.1) satisfies estimate (3.3). □
Theorem 3.3.
Under the conditions of Theorem 3.1 the solution of problem ( 1.1 ) is unique.
Proof.
Let $u_{1}$ , $u_{2}$ be two weak solutions of problem (1.1). Denote $w = u_{2} - u_{1}$ and test (3.1) with the function $ϕ = w$ . This is an admissible choice because $w \in W (Q_{T})$ , $w_{t} \in W^{*} (Q_{T})$ and the space $C^{1} ([0, T]; C_{0}^{1} (Ω))$ is dense in $W (Q_{T}) \cap L_{A}^{q} (Q_{T})$ . For every $t \in [0, T]$ we obtain $\begin{array}{l} \frac{1}{2} \int_{Ω} w^{2} (t) d x + \int_{Q_{t}} D (| \nabla u_{2} |^{p - 2} \nabla u_{2} - | \nabla u_{1} |^{p - 2} \nabla u_{1}) \nabla w d z \\ (3.8) & + \int_{Q_{t}} A (| u_{2} |^{q - 2} u_{2} - | u_{1} |^{q - 2} u_{1}) w d z ⩽ L \int_{Q_{t}} | w |^{2} d z . \end{array}$ Set $Y (t) = ‖ w ‖_{2, Q_{t}}^{2}$ . Because of the well-known inequalities $\forall ξ, η \in R^{n}$ and $r > 1$ $\begin{matrix} (3.9) & (| ξ |^{r - 2} ξ - | η |^{r - 2} η) (ξ - η) ⩾ \{\begin{matrix} 2^{- r} | ξ - η |^{r} & if r ⩾ 2, \\ (r - 1) \frac{| ξ - η |^{2}}{{(| ξ |^{r} + | η |^{r})}^{\frac{2 - r}{r}}} & if r \in (1, 2) \end{matrix} \end{matrix}$ the last two terms on the left-hand side of (3.8) are nonnegative and can be dropped, which leads to the following inequality for $Y (t)$ : $\begin{matrix} Y^{'} (t) ⩽ 2 L Y (t) for t \in (0, T), Y (0) = 0 . \end{matrix}$ By Gronwall’s Lemma $Y (t) ⩽ 0$ for all $t > 0$ , which is only possible if $Y (t) = 0$ for a.e. $t \in (0, T)$ . Since $w \in C ([0, T]; L^{2} (Ω))$ , it follows that $‖ w (t) ‖_{2, Ω} = 0$ for all $t \in [0, T]$ . □

4. Existence of a global attractor in $L^{2} (Ω)$

In this section we derive sufficient conditions for the existence of a global attractor for problem (1.1) in $L^{2} (Ω)$ . We begin by recalling some definitions and results from the nonlinear semigroup theory.

Definition 4.1.

Let $(X, d)$ be a complete metric space. A semigroup is a family ${T (t)}_{t ⩾ 0}$ of single-valued continuous operators $T (t) : X \to X$ depending on a parameter $t \in R^{+}$ and enjoying the semigroup property: $\begin{matrix} T (t_{1}) T (t_{2}) (x) = T (t_{1} + t_{2}) (x), for all t_{1}, t_{2} \in R^{+} and x \in X \end{matrix}$ with $T (0) = I_{d}$ .

Let A and M be subsets of X. We say that A attracts M or M is attracted to A by the semigroup ${T (t)}_{t ⩾ 0}$ if for every $ϵ > 0$ there exists a $t_{1} (ϵ, M) \in R^{+}$ such that $T (t) M \subset O_{ϵ} (A) : = {x \in X; d (x, A) < ϵ}$ for all $t ⩾ t_{1} (ϵ, M)$ .

A is called a global B-attractor if A attracts each bounded set in X.

A semigroup is called bounded dissipative or B-dissipative if it has a bounded global B-attractor.

A set $A \subset X$ is called invariant (relative to the semigroup ${T (t)}$ ) if $T (t) A = A$ , for all $t \in R^{+}$ .

A semigroup ${T (t)}_{t ⩾ 0}$ belongs to the class $K$ if for each $t > 0$ the operator $T (t)$ is compact, i.e., for any bounded set $B \subset X$ its image $T (t) B$ is precompact.

Theorem 4.1 (Theorem 2.2, [14]).

Let ${T (t) : X \to X, t ⩾ 0}$ be a semigroup of class $K$ . If it is B-dissipative, then ${T (t) : X \to X, t ⩾ 0}$ has a minimal closed global B-attractor $M$ , which is compact and invariant.

The main result of this section is given in the following theorem.

Theorem 4.2.
Let $u_{0} \in L^{2} (Ω)$ and f satisfies condition (H3) with $f_{0} \in L^{\infty} (0, T; L^{2} (Ω))$ . Assume that the exponents $p (x)$ , $q (x)$ and the coefficients $A (x)$ , $D (x)$ satisfy conditions (H1)–(H2) and one of the following conditions: $\begin{array}{l} (4.1) & q^{-} > 1, p^{-} > s > 2, D^{- \frac{s}{p (x) - s}} (x) \in L^{1} (Ω); \\ (4.2) & p^{-} > s > \frac{2 n}{n + 2}, q^{-} > 2, A^{\frac{- 2}{q (x) - 2}} (x) \in L^{1} (Ω); \\ (4.3) & \begin{matrix} p^{-} > s > \frac{2 n}{n + 2}, q^{-} > 1, D^{- \frac{s}{p (x) - s}} (x), A^{\frac{- 1}{q (x) - 1}} (x) \in L^{1} (Ω), \\ \frac{2}{q^{-}} + \frac{\frac{1}{p^{-}} - \frac{1}{q^{-}}}{1 + \frac{1}{n} - \frac{1}{s}} < 1 . \end{matrix} \end{array}$ Then the semigroup ${T (t)}$ associated with problem ( 1.1 ) on $L^{2} (Ω)$ has a minimal closed global B-attractor $A$ in $L^{2} (Ω)$ , which is compact and invariant.

Under the conditions of Theorem 4.2 problem (1.1) has a unique global in time weak solution and the proof is reduced to checking the fulfillment of conditions of the abstract Theorem 4.1.
Lemma 4.1.
Let the conditions of Theorem 4.2 be fulfilled. If u is a global weak solution of problem ( 1.1 ) with $u_{0} \in L^{2} (Ω)$ and f satisfying condition (H3), then for every $T > 0$ $\begin{matrix} (4.4) & ‖ u ‖_{L^{\infty} (0, T; L^{2} (Ω))} + \int_{0}^{T} {‖ \nabla u (t) ‖}_{s, Ω}^{s} d t ⩽ C \end{matrix}$ with a positive constant C depending only on the data.
Proof.
The following inequality follows from (3.2) with the use of (H3): for every $t \in (0, T)$ $\begin{array}{l} \frac{1}{2} {‖ u (t) ‖}_{2, Ω}^{2} + \int_{0}^{t} \int_{Ω} D (x) {| \nabla u (s) |}^{p (x)} d x d s + \int_{0}^{t} \int_{Ω} A (x) {| u (s) |}^{q (x)} d x d s \\ (4.5) & ⩽ \frac{1}{2} (‖ u_{0} ‖_{2, Ω}^{2} + ‖ f_{0} ‖_{2, Q_{T}}^{2}) + (L + \frac{1}{2}) \int_{0}^{t} {‖ u (s) ‖}_{2, Ω}^{2} d s . \end{array}$ Dropping the nonnegative terms on the left-hand side of (4.5) we find that for every $t \in (0, T)$ $\begin{matrix} \frac{1}{2} {‖ u (t) ‖}_{2, Ω}^{2} & ⩽ \frac{1}{2} ‖ u_{0} ‖_{2, Ω}^{2} + \frac{1}{2} ‖ f_{0} ‖_{2, Q_{T}}^{2} + (L + \frac{1}{2}) \int_{0}^{t} {‖ u (s) ‖}_{2, Ω}^{2} d s . \end{matrix}$ By the Gronwall–Bellman inequality $\begin{matrix} {‖ u (t) ‖}_{2, Ω}^{2} ⩽ (‖ u_{0} ‖_{2, Ω}^{2} + ‖ f_{0} ‖_{2, Q_{T}}^{2}) e^{(2 L + 1) T} \end{matrix}$ and (4.5) takes the form $\begin{matrix} \frac{1}{2} {‖ u (t) ‖}_{2, Ω}^{2} + \int_{0}^{t} \int_{Ω} D (x) {| \nabla u (s) |}^{p (x)} d x d s ⩽ (\frac{1}{2} + T (\frac{1}{2} + L) e^{(2 L + 1) T}) (‖ u_{0} ‖_{2, Ω}^{2} + ‖ f_{0} ‖_{2, Q_{T}}^{2}) . \end{matrix}$ By (2.10) $\begin{matrix} \int_{0}^{T} {‖ \nabla u (t) ‖}_{s, Ω}^{s} d t ⩽ \int_{0}^{T} \int_{Ω} D {| \nabla u (t) |}^{p} d x d t + T \int_{Ω} D^{- \frac{s}{p - s}} d x \end{matrix}$ and (4.4) follows. □
Lemma 4.2.

If $u_{0}, v_{0} \in L^{2} (Ω)$ , $u (\cdot) : = T (\cdot) u_{0}$ and $v (\cdot) : = T (\cdot) v_{0}$ , then $\begin{matrix} {‖ u (t) - v (t) ‖}_{2, Ω} ⩽ ‖ u_{0} - v_{0} ‖_{2, Ω} e^{2 L T}, \forall t \in [0, T] . \end{matrix}$

The map $T : R^{+} \times L^{2} (Ω) \to L^{2} (Ω)$ is continuous.

Proof.
(i) Let u, v be two weak solutions of problem (1.1). As in the proof of Theorem 3.3 for the difference $w = u - v$ we obtain equality (3.8), whence $\begin{matrix} \frac{1}{2} {‖ u (t) - v (t) ‖}_{2, Ω}^{2} ⩽ \frac{1}{2} ‖ u_{0} - v_{0} ‖_{2, Ω}^{2} + L \int_{0}^{t} {‖ u (s) - v (s) ‖}_{2, Ω}^{2} d s \end{matrix}$ and the result follows by Gronwall’s Lemma.

(ii) Let us take $(t_{0}, v_{0}) \in [0, T] \times L^{2} (Ω)$ . Since $v \in C ([0, T]; L^{2} (Ω))$ , for every $ϵ > 0$ there is $δ_{1} > 0$ such that $\begin{matrix} {‖ T (t) v_{0} - T (t_{0}) v_{0} ‖}_{2, Ω} < \frac{ϵ}{2} \end{matrix}$ whenever $| t - t_{0} | < δ_{1}$ . Let us take $δ : = min {\frac{ϵ}{2} e^{- 2 L T}, δ_{1}} > 0$ . Using item (i) we have: $\begin{matrix} {‖ T (t) u_{0} - T (t_{0}) v_{0} ‖}_{2, Ω} & ⩽ {‖ T (t) u_{0} - T (t) v_{0} ‖}_{2, Ω} + {‖ T (t) v_{0} - T (t_{0}) v_{0} ‖}_{2, Ω} \\ ⩽ e^{2 L T} ‖ u_{0} - v_{0} ‖_{2, Ω} + \frac{ϵ}{2} < ϵ, \end{matrix}$ provided that $‖ (t, u_{0}) - (t_{0}, v_{0}) ‖_{[0, T] \times L^{2} (Ω)} < δ$ . □
Lemma 4.3.
Let ${T (t)}$ be the semigroup associated with problem ( 1.1 ) on $L^{2} (Ω)$ . Then $T (t) : L^{2} (Ω) \to L^{2} (Ω)$ is of class $K$ .
Proof.
We adapt the proof of [17, Theorem 3]. Let ${u_{0 k}}$ be a sequence of initial data, $‖ u_{0 k} ‖_{2, Ω} ⩽ r$ , and $u_{k} (t) = T (t) u_{0 k}$ the corresponding solution of problem (1.1). Take an arbitrary interval $(α, β) \subset (0, T)$ with $2 α < β$ . Applying (2.10) we may estimate $\begin{matrix} \frac{1}{β - α} \int_{α}^{β} {‖ \nabla u_{k} (t) ‖}_{s, Ω}^{s} d t & ⩽ \frac{1}{β - α} \int_{α}^{β} \int_{Ω} D | \nabla u_{k} |^{p} d x d t + \int_{Ω} D^{- \frac{s}{p - s}} d x \\ ⩽ \frac{C}{α} (r^{2} + ‖ f_{0} ‖_{2, Q_{T}}^{2}) + C = : K_{0} . \end{matrix}$ There exists a sequence ${t_{k}} \subset [α, β]$ such that $\begin{matrix} I_{k} (t_{k}) = {‖ \nabla u_{k} (t_{k}) ‖}_{s, Ω}^{s} ⩽ K_{0} + 1 . \end{matrix}$ Indeed: if we assume that $I_{k} (t) > K_{0} + 1$ for some $k \in N$ and all $t \in (α, β)$ , then $\begin{matrix} K_{0} ⩾ \frac{1}{β - α} \int_{α}^{β} {‖ \nabla u_{k} (t) ‖}_{2, Ω}^{s} d t = \frac{1}{β - α} \int_{α}^{β} I_{k} (t) d t ⩾ K_{0} + 1, \end{matrix}$ which is impossible. Since $‖ \nabla u_{k} (t_{k}) ‖_{s, Ω}$ are uniformly bounded and the embedding $W_{0}^{1, s} (Ω) \subset L^{2} (Ω)$ is compact, there is a subsequence ${t_{k_{j}}}$ such that $\begin{matrix} t_{k_{j}} \to t_{0} and T (t_{k_{j}}) u_{0 k_{j}} = u_{k_{j}} (t_{k_{j}}) \to v_{0} in L_{2} (Ω) as k_{j} \to \infty . \end{matrix}$ By Lemma 4.2(ii) the map T is continuous and $T (t) u_{0 k_{j}} = T (t - t_{k_{j}}) T (t_{k_{j}}) u_{0 k_{j}} \to T (t - t_{0}) v_{0}$ as $k_{j} \to \infty$ . □
Lemma 4.4.
Assume that conditions ( 4.1 ) or ( 4.2 ) are fulfilled. For every $u \in W \cap L_{A (\cdot)}^{q (\cdot)} (Ω)$ $\begin{matrix} μ {(‖ u ‖_{2, Ω}^{2})}^{ν} ⩽ \int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x + λ, 1 < ν = \{\begin{matrix} \frac{s}{2} & if (4.1) is fulfilled, \\ \frac{q^{-}}{2} & if (4.2) holds \end{matrix} \end{matrix}$ with an independent of u constant λ and $\begin{matrix} μ = \{\begin{matrix} {(2 ‖ A^{- \frac{2}{q}} ‖_{\frac{q (\cdot)}{q (\cdot) - 2}, Ω})}^{\frac{2}{q^{-}}} & if q^{-} > 2, \\ 1 & if s > 2 . \end{matrix} \end{matrix}$
Proof.
Let $s > 2$ . By the Sobolev embedding theorem and (2.10) $\begin{matrix} {(‖ u ‖_{2, Ω}^{2})}^{\frac{s}{2}} & ⩽ C \int_{Ω} | \nabla u |^{s} d x ⩽ \int_{Ω} D | \nabla u |^{p} d x + C^{'} \int_{Ω} D^{- \frac{s}{p - s}} d x . \end{matrix}$ If $q^{-} > 2$ , by Hölder’s and Young’s inequalities $\begin{matrix} \int_{Ω} u^{2} d x & = \int_{Ω} {(A | u |^{q (x)})}^{\frac{2}{q (x)}} A^{- \frac{2}{q (x)}} d x \\ ⩽ 2 {‖ {(A | u |^{q (x)})}^{\frac{2}{q (x)}} ‖}_{\frac{q (\cdot)}{2}, Ω} {‖ A^{- \frac{2}{q (x)}} ‖}_{\frac{q (\cdot)}{q (\cdot) - 2}, Ω} \\ ⩽ 2 {‖ A^{- \frac{2}{q (x)}} ‖}_{\frac{q (\cdot)}{q (\cdot) - 2}, Ω} max {{(\int_{Ω} A | u |^{q} d x)}^{\frac{2}{q^{-}}}, {(\int_{Ω} A | u |^{q} d x)}^{\frac{2}{q^{+}}}} \\ ⩽ 2 {‖ A^{- \frac{2}{q (x)}} ‖}_{\frac{q (\cdot)}{q (\cdot) - 2}, Ω} {(1 + \int_{Ω} A | u |^{q} d x)}^{\frac{2}{q^{-}}} \\ ⩽ 2 {‖ A^{- \frac{2}{q (x)}} ‖}_{\frac{q (\cdot)}{q (\cdot) - 2}, Ω} {(1 + \int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x)}^{\frac{2}{q^{-}}} . \end{matrix}$ □
Lemma 4.5.
Let conditions ( 4.3 ) be fulfilled. For every $u \in W \cap L_{A (\cdot)}^{q (\cdot)} (Ω)$ $\begin{matrix} (4.6) & μ {(‖ u ‖_{2, Ω}^{2})}^{ν} ⩽ \int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x + 1, \frac{1}{ν} = \frac{2}{q^{-}} + \frac{\frac{1}{p^{-}} - \frac{1}{q^{-}}}{1 + \frac{1}{n} - \frac{1}{s}} \end{matrix}$ and an independent of u constant μ.
Proof.
Set $\begin{matrix} \frac{1}{α} = 2 (1 + \frac{1}{n} - \frac{1}{s}) . \end{matrix}$ It is straightforward to check that $α \in (0, 1)$ , provided that $s > \frac{2 n}{n + 2}$ . Let us make use of the Gagliardo–Nirenberg inequality: $\begin{matrix} (4.7) & ‖ u ‖_{2, Ω} ⩽ C_{} ‖ \nabla u ‖_{s, Ω}^{α} ‖ u ‖_{1, Ω}^{1 - α} \forall u \in W_{0}^{1, s} (Ω) \end{matrix}$ with an independent of u constant $C_{}$ . By the generalized Hölder inequality (2.1) and (2.2) $\begin{array}{l} ‖ \nabla u ‖_{s, Ω}^{α} & = {(\int_{Ω} (D^{\frac{s}{p}} | \nabla u |^{s}) (D^{- \frac{s}{p}}) d x)}^{\frac{α}{s}} ⩽ 2^{α} {‖ D^{\frac{s}{p}} | \nabla u |^{s} ‖}_{\frac{p (\cdot)}{s}, Ω}^{α} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α} \\ ⩽ 2^{α} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α} max {{(\int_{Ω} D | \nabla u |^{p} d x)}^{\frac{α}{p^{+}}}, {(\int_{Ω} D | \nabla u |^{p} d x)}^{\frac{α}{p^{-}}}} \\ ⩽ 2^{α} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α} {(1 + \int_{Ω} D | \nabla u |^{p} d x)}^{\frac{α}{p^{-}}}, \\ ‖ u ‖_{1, Ω}^{1 - α} & = {(\int_{Ω} {(A | u |^{q})}^{\frac{1}{q}} A^{- \frac{1}{q}} d x)}^{1 - α} ⩽ 2^{1 - α} {‖ A | u |^{q} ‖}_{q (\cdot), Ω}^{1 - α} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} \\ ⩽ 2^{1 - α} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} max {{(\int_{Ω} A | u |^{q} d x)}^{\frac{1 - α}{q^{+}}}, {(\int_{Ω} A | u |^{q} d x)}^{\frac{1 - α}{q^{-}}}} \\ ⩽ 2^{1 - α} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} {(1 + \int_{Ω} A | u |^{q} d x)}^{\frac{1 - α}{q^{-}}} . \end{array}$ Plugging these estimates into (4.7) we arrive at the inequality $\begin{matrix} ‖ u ‖_{2, Ω} & ⩽ 2 C_{} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} {(1 + \int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x)}^{\frac{α}{p^{-}} + \frac{1 - α}{q^{-}}}, \end{matrix}$ which is equivalent to (4.6) with the constant $\begin{matrix} \frac{1}{μ} = {(2 C_{} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α})}^{\frac{1}{q^{-}} + α (\frac{1}{p^{-}} - \frac{1}{q^{-}})} . \end{matrix}$ □
Lemma 4.6.
Let A, D, p, q satisfy the conditions of Lemma 4.5 . For every $u \in W \cap L_{A (\cdot)}^{q (\cdot)} (Ω)$ with $‖ u ‖_{2, Ω}^{2} ⩽ M$ $\begin{matrix} (4.8) & γ min {M^{σ^{-} - σ^{+}}, 1} {(‖ u ‖_{2, Ω}^{2})}^{σ^{+}} ⩽ \int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x, \frac{1}{σ^{\pm}} = \frac{2}{q^{\pm}} + \frac{\frac{1}{p^{\pm}} - \frac{1}{q^{\pm}}}{1 + \frac{1}{n} - \frac{1}{s}}, \end{matrix}$ with the constants $C_{}$ and* $α \in (0, 1)$ from ( 4.7 ) and $\begin{matrix} γ = min {δ^{σ^{+}}, δ^{σ^{-}}}, \frac{1}{δ} = 2 C_{} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α} . \end{matrix}$
Proof.
Let us denote $\begin{matrix} Λ = \int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x . \end{matrix}$ Following the proof of Lemma 4.5 we obtain the inequality $\begin{matrix} ‖ u ‖_{2, Ω}^{2} & ⩽ 2 C_{} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α} \\ \times max {{(\int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x)}^{\frac{α}{p^{-}} + \frac{1 - α}{q^{-}}}, {(\int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x)}^{\frac{α}{p^{+}} + \frac{1 - α}{q^{+}}}} \\ = \frac{1}{δ} \{\begin{matrix} Λ^{\frac{α}{p^{+}} + \frac{1 - α}{q^{+}}} & if Λ ⩽ 1, \\ Λ^{\frac{α}{p^{-}} + \frac{1 - α}{q^{-}}} & if Λ > 1, \end{matrix} \frac{1}{σ^{\pm}} = \frac{α}{p^{\pm}} + \frac{1 - α}{q^{\pm}} . \end{matrix}$ The last inequality leads to the chain of relations $\begin{array}{l} γ min {{(‖ u ‖_{2, Ω}^{2})}^{σ^{+}}, {(‖ u ‖_{2, Ω}^{2})}^{σ^{-}}} & = min {δ^{σ^{+}}, δ^{σ^{-}}} min {{(‖ u ‖_{2, Ω}^{2})}^{σ^{+}}, {(‖ u ‖_{2, Ω}^{2})}^{σ^{-}}} \\ ⩽ min {{(δ ‖ u ‖_{2, Ω}^{2})}^{σ^{+}}, {(δ ‖ u ‖_{2, Ω}^{2})}^{σ^{-}}} \\ (4.9) & ⩽ Λ = \int_{Ω} (D | \nabla u |^{p} + A | u |^{q}) d x . \end{array}$ Notice that since $‖ u ‖_{2, Ω}^{2} ⩽ M$ by assumption, we have $\begin{array}{l} γ min {1, M^{σ^{-} - σ^{+}}} {(‖ u ‖_{2, Ω}^{2})}^{σ^{+}} \\ = γ min {M^{σ^{+}}, M^{σ^{-}}} {(\frac{‖ u ‖_{2, Ω}^{2}}{M})}^{σ^{+}} \\ ⩽ γ min {M^{σ^{+}}, M^{σ^{-}}} min {{(\frac{‖ u ‖_{2, Ω}^{2}}{M})}^{σ^{+}}, {(\frac{‖ u ‖_{2, Ω}^{2}}{M})}^{σ^{-}}} \\ ⩽ γ min {M^{σ^{+}} {(\frac{‖ u ‖_{2, Ω}^{2}}{M})}^{σ^{+}}, M^{σ^{-}} {(\frac{‖ u ‖_{2, Ω}^{2}}{M})}^{σ^{-}}} \\ (4.10) & = γ min {{(‖ u ‖_{2, Ω}^{2})}^{σ^{+}}, {(‖ u ‖_{2, Ω}^{2})}^{σ^{-}}} . \end{array}$ Gathering (4.9) with (4.10) we obtain (4.8). □
Lemma 4.7.
Let $u_{0} \in L^{2} (Ω)$ and $f \in L^{\infty} (0, T; L^{2} (Ω))$ . Under the conditions of Theorem 4.2 the semigroup ${T (t)}$ associated with problem ( 1.1 ) on $L^{2} (Ω)$ is bounded dissipative in $L^{2} (Ω)$ .
Proof.
Let us fix some $T > 0$ and $t, t + h \in (0, T)$ . Subtracting identities (3.2) with $τ = t + h$ and $t = τ$ and dividing the result by h we arrive at the equality $\begin{array}{l} \frac{1}{2 h} {‖ u (τ) ‖}_{2, Ω}^{2} |_{τ = t}^{τ = t + h} \\ = - \frac{1}{h} \int_{t}^{t + h} \int_{Ω} D | \nabla u |^{p} d z - \frac{1}{h} \int_{t}^{t + h} \int_{Ω} A | u |^{q} d z + \frac{1}{h} \int_{t}^{t + h} \int_{Ω} f (x, t, u) u d z . \end{array}$ Since $u \in W (Q_{T})$ and $A | u |^{q} \in L^{1} (Q_{T})$ , by the Lebesgue differentiation theorem each term on the right-hand side of this equality has a limit as $h \to 0$ for a.e. $t \in (0, T)$ . Letting $h \to 0$ we get $\begin{matrix} (4.11) & \frac{1}{2} \frac{d}{d t} ({‖ u (t) ‖}_{2, Ω}^{2}) + \int_{Ω} D (x) | \nabla u |^{p (x)} d x + \int_{Ω} A (x) | u |^{q (x)} d x = \int_{Ω} f (x, t, u) u d x \end{matrix}$ for a.e. $t \in (0, T)$ . Equality (4.11) leads to a differential inequality for the function $‖ u (t) ‖_{2, Ω}^{2}$ , which we derive separately in the cases (4.1), (4.2) and (4.3).

(i) If (4.1) holds, by (4.11), Lemma 4.4, assumptions (H3) and Young’s inequality $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ({‖ u (t) ‖}_{2, Ω}^{2}) + {({‖ u (t) ‖}_{2, Ω}^{2})}^{\frac{s}{2}} \\ ⩽ L {‖ u (t) ‖}_{2, Ω}^{2} + \int_{Ω} | f_{0} (t) | | u (t) | d x + λ \\ ⩽ (\frac{1}{2} + L) {‖ u (t) ‖}_{2, Ω}^{2} + \frac{1}{2} ess sup_{(0, T)} {‖ f_{0} (t) ‖}_{2, Ω}^{2} + λ \\ ⩽ \frac{1}{2} {({‖ u (t) ‖}_{2, Ω}^{2})}^{\frac{s}{2}} + \frac{1}{2} {(1 + 2 L)}^{\frac{s}{s - 2}} + \frac{1}{2} ess sup_{(0, T)} ‖ f_{0} ‖_{2, Ω}^{2} + λ . \end{array}$ It follows that $y (t) = ‖ u (t) ‖_{2, Ω}^{2}$ satisfies the ordinary differential inequality $\begin{matrix} y^{'} (t) + y^{ν} (t) ⩽ M for a.e. t \in (0, T) \end{matrix}$ with the constants $\begin{matrix} ν = \frac{s}{2} > 1, M = {(1 + 2 L)}^{\frac{s}{s - 2}} + ess sup_{(0, T)} ‖ f_{0} ‖_{2, Ω}^{2} + λ . \end{matrix}$

(ii) If (4.2) holds, then $q^{-} > 2$ . By Lemma 4.4 and Young’s inequality $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ({‖ u (t) ‖}_{2, Ω}^{2}) + μ {({‖ u (t) ‖}_{2, Ω}^{2})}^{\frac{q^{-}}{2}} \\ ⩽ (\frac{1}{2} + L) {‖ u (t) ‖}_{2, Ω}^{2} + \frac{1}{2} ess sup_{(0, T)} ‖ f_{0} ‖_{2, Ω}^{2} + λ \\ ⩽ \frac{μ}{2} {({‖ u (t) ‖}_{2, Ω}^{2})}^{\frac{q^{-}}{2}} + \frac{1}{2} {(1 + 2 L)}^{\frac{q^{-}}{q^{-} - 2}} μ^{\frac{q^{-}}{q^{-} - 2}} + \frac{1}{2} ess sup_{(0, T)} ‖ f_{0} ‖_{2, Ω}^{2} + λ . \end{array}$ This leads to the ordinary differential inequality for the function $y (t)$ : for a.e. $t \in (0, T)$ $\begin{matrix} y^{'} (t) + μ y^{\frac{q^{-}}{2}} (t) ⩽ M^{'}, M^{'} = 2 λ + {(1 + 2 L)}^{\frac{q^{-}}{q^{-} - 2}} μ^{\frac{q^{-}}{q^{-} - 2}} + ess sup_{(0, T)} ‖ f_{0} ‖_{2, Ω}^{2} . \end{matrix}$

(iii) Let condition (4.3) be fulfilled. Arguing as in the case (i), from (4.11) and Lemma 4.5 we obtain the following inequality: $\begin{matrix} \frac{1}{2} y^{'} (t) & + μ y^{ν} (t) ⩽ L y (t) + \int_{Ω} | f_{0} | | u (t) | d x + 1 ⩽ \frac{μ}{2} y^{ν} (t) + \frac{1}{2} ess sup_{(0, T)} ‖ f_{0} ‖_{2, Ω}^{2} + C (μ, ν, L) . \end{matrix}$

Thus, in all three cases $y (t)$ satisfies the inequality $\begin{matrix} (4.12) & y^{'} (t) + C_{0} y^{θ} (t) ⩽ C_{1} a.e. in (0, T), y (0) = ‖ u_{0} ‖_{2, Ω}^{2} \end{matrix}$ with some $θ > 1$ and independent of $u_{0}$ constants $C_{0}$ , $C_{1}$ . By [22, Lemma 5.1, p. 163], if a function $y (t)$ satisfies inequality (4.12), then for every $t_{0} \in (0, T)$ and all $t > t_{0}$ $\begin{matrix} y (t) = {‖ u (t) ‖}_{2, Ω}^{2} ⩽ {(C_{1} C_{0})}^{\frac{2}{θ}} + {(2 C_{0} (2 θ - 2) t_{0})}^{\frac{- 1}{2 (θ - 1)}} = : D . \end{matrix}$ In every case the set ${ω \in L^{2} (Ω) : ‖ ω ‖_{2, Ω}^{2} ⩽ D}$ attracts bounded sets of $L^{2} (Ω)$ in the $L^{2}$ -norm. □

By Lemmas 4.3, 4.7 the semigroup ${T (t)}$ is of class $K$ and bounded-dissipative in $L^{2} (Ω)$ . The assertion of Theorem 4.2 follows from Theorem 4.1.
5. Vanishing in a finite time and exponential decay

Let us denote by $C_{s}$ and $C_{*}$ the constants in the Sobolev embedding inequality $\begin{matrix} (5.1) & ‖ v ‖_{2, Ω} ⩽ C_{s} ‖ \nabla v ‖_{s, Ω}, v \in W_{0}^{1, s} (Ω), \end{matrix}$ and the Gagliardo–Nirenberg inequality (4.7).

Theorem 5.1.

Let u be a nontrivial strong solution of problem ( 1.1 ). Assume that the exponents p, q and the coefficients D, A satisfy conditions (H1), (H2) and $\begin{matrix} (5.2) & \begin{matrix} (a) & \frac{2 n}{n + 2} < s < p^{-} ⩽ p^{+} < 2, \\ condition (H3) is fulfilled with f_{0} = 0 and L > 0, \\ (b) & ess sup_{(0, T)} {‖ u (t) ‖}_{2, Ω}^{2} = M, M = const, \\ (c) & ‖ u_{0} ‖_{2, Ω}^{2 - p^{+}} < \frac{K}{L} \\ with K = min {μ^{- p^{+}}, μ^{- p^{-}}} min {1, M^{p^{-} - p^{+}}}, μ = 2^{\frac{1}{s}} C_{s} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{\frac{1}{s}} . \end{matrix} \end{matrix}$ Then $u (z)$ vanishes at the finite moment: $\begin{matrix} {‖ u (t) ‖}_{2, Ω} = 0 for all t ⩾ t^{*} = \frac{- 1}{L (2 - p^{+})} ln (1 - \frac{L}{K} ‖ u_{0} ‖_{2, Ω}^{2 - p^{+}}) . \end{matrix}$

Proof.

Let u be a nontrivial strong solution of problem (1.1). Let us denote $\begin{matrix} Σ (t) = \int_{Ω} D {| \nabla u (t) |}^{p} d x . \end{matrix}$ Using inequality (5.1) and then arguing as in the proof of Lemma 4.6 we find that for a.e. $t \in (0, T)$ $\begin{array}{l} {‖ u (t) ‖}_{2, Ω} & ⩽ C_{s} {‖ \nabla u (t) ‖}_{s, Ω} ⩽ 2^{\frac{1}{s}} C_{s} {‖ D^{\frac{s}{p}} {| \nabla u (t) |}^{s} ‖}_{\frac{p (\cdot)}{s}, Ω}^{\frac{1}{s}} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{\frac{1}{s}} \\ ⩽ 2^{\frac{1}{s}} C_{s} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{\frac{1}{s}} max {{(\int_{Ω} D {| \nabla u (t) |}^{p} d x)}^{\frac{1}{p^{+}}}, {(\int_{Ω} D {| \nabla u (t) |}^{p} d x)}^{\frac{1}{p^{-}}}} \\ (5.3) & = μ \{\begin{matrix} Σ^{\frac{1}{p^{+}}} (t) & if Σ (t) ⩽ 1, \\ Σ^{\frac{1}{p^{-}}} (t) & if Σ (t) > 1, \end{matrix} \end{array}$ which is equivalent to the inequality $\begin{matrix} min {{(μ^{- 1} {‖ u (t) ‖}_{2, Ω})}^{p^{+}}, {(μ^{- 1} {‖ u (t) ‖}_{2, Ω})}^{p^{-}}} ⩽ Σ (t) . \end{matrix}$ The following chain of relations holds (see the derivation of (4.10)): $\begin{array}{l} K {({‖ u (t) ‖}_{2, Ω}^{2})}^{\frac{p^{+}}{2}} & = min {μ^{- p^{+}}, μ^{- p^{-}}} min {M^{p^{+}}, M^{p^{-}}} {(\frac{‖ u (t) ‖_{2, Ω}^{2}}{M^{2}})}^{\frac{p^{+}}{2}} \\ ⩽ min {μ^{- p^{+}}, μ^{- p^{-}}} min {{({‖ u (t) ‖}_{2, Ω}^{2})}^{\frac{p^{+}}{2}}, {({‖ u (t) ‖}_{2, Ω}^{2})}^{\frac{p^{-}}{2}}} \\ ⩽ min {{(μ^{- 1} {‖ u (t) ‖}_{2, Ω})}^{p^{+}}, {(μ^{- 1} {‖ u (t) ‖}_{2, Ω})}^{p^{-}}} \\ (5.4) & ⩽ Σ (t) = \int_{Ω} D {| \nabla u (t) |}^{p} d x . \end{array}$ Plugging this estimate into inequality (4.11) for the solution $u (z)$ and using condition (1.3) with $f_{0} = 0$ and $L > 0$ , we arrive at the differential inequality for the function $y (t) = ‖ u (t) ‖_{2, Ω}^{2}$ , $\begin{matrix} (5.5) & y^{'} (t) + 2 K y^{ν} (t) ⩽ 2 L y (t) a.e. in (0, T), y (0) = ‖ u_{0} ‖_{2, Ω}^{2}, ν = \frac{p^{+}}{2} < 1, \end{matrix}$ which is transformed into the inequality for $z (t) = y (t) e^{- 2 L t}$ : $\begin{matrix} (5.6) & z^{'} (t) + 2 K e^{- 2 (1 - ν) L t} z^{ν} (t) ⩽ 0 a.e. in (0, T), z (0) = ‖ u_{0} ‖_{2, Ω}^{2}, 2 (1 - ν) = 2 - p^{+} . \end{matrix}$ Let us denote $\tilde{t} = sup {t \in (0, T) : ‖ u (t) ‖_{2, Ω} > 0}$ . Notice that $\tilde{t} > 0$ because $u \in C ([0, T]; L^{2} (Ω))$ is a nontrivial solution of problem (1.1). Multiplying inequality (5.6) by $z^{- ν} (t)$ and integrating over the interval $[0, t) \subset [0, \tilde{t})$ we arrive at the inequality $\begin{matrix} 0 ⩽ z^{1 - ν} (t) ⩽ ‖ u_{0} ‖_{2, Ω}^{2 - p^{+}} - \frac{K}{L} (1 - e^{- (2 - p^{+}) L t}) = : Φ (t) . \end{matrix}$ Since $Φ (0) = ‖ u_{0} ‖_{2, Ω}^{2 - p^{+}} > 0$ , $Φ (\infty) < 0$ by assumption, and $Φ (t)$ is monotone decreasing, it is necessary that $Φ (t) ⩽ 0$ for all $t ⩾ t^{*}$ , which means that $‖ u (t) ‖_{2, Ω}^{2} \equiv 0$ for all $t ⩾ t^{*}$ . □

Let us recall the notations from Lemma 4.6: $\begin{matrix} (5.7) & \begin{matrix} \frac{1}{α} = 2 (1 + \frac{1}{n} - \frac{1}{s}), \frac{1}{δ} = 2 C_{*} {‖ A^{- \frac{1}{q}} ‖}_{q^{'} (\cdot), Ω}^{1 - α} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{α}, \\ \frac{1}{σ^{\pm}} = \frac{2}{q^{\pm}} + \frac{\frac{1}{p^{\pm}} - \frac{1}{q^{\pm}}}{1 + \frac{1}{n} - \frac{1}{s}}, γ = min {δ^{σ^{+}}, δ^{σ^{-}}} . \end{matrix} \end{matrix}$

Theorem 5.2.

Let $u (z)$ be a nontrivial strong solution of problem ( 1.1 ) such that $\begin{matrix} ess sup_{(0, T)} {‖ u (t) ‖}_{2, Ω}^{2} = M, M = const . \end{matrix}$ Assume that the exponents p, q and the coefficients D, A satisfy conditions (H1), (H2) and $\begin{matrix} (5.8) & \begin{matrix} p^{-} > s > \frac{2 n}{n + 2}, q^{-} > 1, A^{\frac{- 1}{q (x) - 1}} (x) \in L^{1} (Ω), \frac{1}{σ^{+}} > 1, \\ ‖ u_{0} ‖_{2, Ω}^{2 (1 - σ^{+})} < \frac{K}{L} with K = γ min {1, M^{σ^{-} - σ^{+}}} . \end{matrix} \end{matrix}$ Then $u (z)$ vanishes at the finite moment: $\begin{matrix} {‖ u (t) ‖}_{2, Ω} = 0 for all t ⩾ t^{*} = \frac{- 1}{2 L (1 - σ^{+})} ln (1 - \frac{L}{K} ‖ u_{0} ‖_{2, Ω}^{2 (1 - σ^{+})}) . \end{matrix}$

Proof.

Under the conditions of Lemma 4.6 the strong solutions of problem (1.1) satisfy inequality (4.8). Plugging it into (4.11) and using (H3) with $f_{0} = 0$ we obtain the differential inequality for $y (t) = ‖ u (t) ‖_{2, Ω}^{2}$ : $\begin{matrix} (5.9) & y^{'} (t) + 2 K y^{σ^{+}} (t) ⩽ 2 L y (t) for a.e. t \in (0, T), y (0) = ‖ u_{0} ‖_{2, Ω}^{2} . \end{matrix}$ The proof is concluded as in Theorem 5.1. □

Corollary 5.1.

Let in the conditions of Theorems 5.1 or 5.2 $L = 0$ , i.e., $f (x, t, u) \equiv 0$ . Then every strong solution of problem ( 1.1 ) with $u_{0} \in L^{2} (Ω)$ vanishes in a finite time: $\begin{matrix} (5.10) & {‖ u (t) ‖}_{2, Ω} = 0 for all t ⩾ t^{*} = \frac{‖ u_{0} ‖_{2, Ω}^{2 (1 - ν)}}{2 K (1 - ν)}, ν = \{\begin{matrix} \frac{p^{+}}{2} & in Theorem 5.1, \\ σ^{+} & in Theorem 5.2 \end{matrix} \end{matrix}$ with the constant K defined in ( 5.2 ) and ( 5.8 ).

Proof.

In the case $L = 0$ we obtain the inequalities $\begin{matrix} y^{'} (t) + 2 K y^{ν} (t) ⩽ 0 \end{matrix}$ with the exponent $ν < 1$ defined in (5.10). The straightforward integration gives the estimate $\begin{matrix} y^{1 - ν} (t) ⩽ ‖ u_{0} ‖_{2, Ω}^{2 (1 - ν)} - 2 K (1 - ν) t \end{matrix}$ and the conclusion follows: $\begin{matrix} {‖ u (t) ‖}_{2, Ω}^{2} = 0 for all t ⩾ t^{*} = \frac{‖ u_{0} ‖_{2, Ω}^{2 (1 - ν)}}{2 K (1 - ν)} . \end{matrix}$ □

Theorem 5.3.

Let us assume that either the conditions of Theorem 5.1 are fulfilled with $p^{+} = 2$ , or the conditions of Theorem 5.2 hold with $σ^{+} = 1$ . If the constant K defined in ( 5.2 ) (Theorem 5.1 ) and ( 5.8 ) (Theorem 5.2 ) satisfies the inequality $K > L$ , then the strong solution of problem ( 1.1 ) decreases exponentially as $t \to \infty$ : $\begin{matrix} {‖ u (t) ‖}_{2, Ω}^{2} ⩽ ‖ u_{0} ‖_{2, Ω}^{2} e^{- 2 (K - L) t} for all t ⩾ 0 . \end{matrix}$

Proof.

It is sufficient to consider the case $p^{+} = 2$ , the case $σ^{+} = 1$ is studied in the same way. Under the conditions of Theorem 5.1 $y (t) = ‖ u (t) ‖_{2, Ω}^{2}$ satisfies the inequality $\begin{matrix} y^{'} (t) + 2 (K - L) y (t) ⩽ 0 a.e. in (0, T), y (0) = ‖ u_{0} ‖_{2, Ω}^{2}, \end{matrix}$ which can be immediately integrated: $\begin{matrix} {‖ u (t) ‖}_{2, Ω}^{2} ⩽ ‖ u_{0} ‖_{2, Ω}^{2} e^{- 2 (K - L) t} . \end{matrix}$ □

The only estimate on the constant M in conditions (5.2), (5.8) is given in the proof of the existence theorem. Although this estimate depends on T and may grow as $T \to \infty$ , in the conditions of Theorems 5.1, 5.2 the value of the constant M is assumed to be a priori known. One may find examples of the situation in which the constant M can be estimated independently of T. Let us assume, for example, that $f (x, t, u) = a (t) u$ where $| a (t) | ⩽ L$ and $\begin{matrix} | \int_{0}^{\infty} a (t) d t | = μ . \end{matrix}$ It follows from (4.11) that $y (t) = ‖ u (t) ‖_{2, Ω}^{2}$ satisfies the inequality $\begin{matrix} y^{'} (t) ⩽ 2 a (t) y (t) a.e. in (0, T), y (0) = ‖ u_{0} ‖_{2, Ω}^{2}, \end{matrix}$ whence, by Gronwall’s inequality, $y (t) ⩽ y (0) exp (2 \int_{0}^{t} a (τ) d τ) ⩽ y (0) e^{2 μ} = M$ . Another example is furnished by the equation with $f (x, t, u) u ⩽ 0$ a.e. in $Q_{T}$ . For such an equation estimate (4.11) takes the form $y^{'} (t) ⩽ 0$ a.e. in $(0, T)$ , which leads to the inequality $M ⩽ ‖ u_{0} ‖_{2, Ω}$ .

The assertions of Theorems 5.1, 5.2 remain true if we omit the assumption that the value of M is given and estimate M through the problem data. In the next two theorems we show that such a change of the conditions does not lead to a qualitative alteration of the conclusions about the long-time behavior of the solution, but may require further quantitative restrictions on the norm of the initial datum.

Theorem 5.4.

Let $u (z)$ be a nontrivial strong solution of problem ( 1.1 ). Assume that the exponents p, q and the coefficients D, A satisfy conditions (H1), (H2) and $\begin{matrix} (5.11) & \begin{matrix} (a) & \frac{2 n}{n + 2} < s < p^{-} ⩽ p^{+} < 2, condition (H3) is fulfilled with f_{0} = 0 and L > 0, \\ (b) & ‖ u_{0} ‖_{2, Ω}^{2 - p^{+}} < min {1, \frac{(2 - p^{+}) κ}{2 L (\frac{p^{+}}{2} - p^{-} + 1)}} \\ with the constants κ = min {μ^{- p^{+}}, μ^{- p^{-}}}, μ = 2^{\frac{1}{s}} C_{s} {‖ D^{- \frac{s}{p}} ‖}_{\frac{p (\cdot)}{p (\cdot) - s}, Ω}^{\frac{1}{s}} . \end{matrix} \end{matrix}$ Then $u (z)$ vanishes at the finite moment: $\begin{matrix} {‖ u (t) ‖}_{2, Ω} = 0 for all t ⩾ t^{*} = \frac{- 1}{2 L (\frac{p^{+}}{2} - p^{-} + 1)} ln (1 - \frac{2 L (\frac{p^{+}}{2} - p^{-} + 1)}{κ (2 - p^{+})} ‖ u_{0} ‖_{2, Ω}^{2 - p^{+}}) . \end{matrix}$

Proof.

Let take $t \in (0, T)$ and repeat the proof of Theorem 5.1 on the interval $(0, t)$ instead of $(0, T)$ . For every strong solution $u (z)$ there is a function $M (t)$ such that $\begin{matrix} (5.12) & ess sup_{(0, t)} {‖ u (τ) ‖}_{2, Ω}^{2} ⩽ M (t), t \in (0, T) . \end{matrix}$ For a.e. $τ \in (0, t)$ the function $u (x, τ)$ satisfies inequalities (5.3) and (5.4) with the constant M substituted by the function $M (t)$ . By virtue of (4.11), the function $y (τ) = ‖ u (τ) ‖_{2, Ω}^{2}$ satisfies the differential inequality $\begin{matrix} (5.13) & y^{'} (τ) + 2 κ min {1, M^{p^{-} - p^{+}} (t)} y^{\frac{p^{+}}{2}} (τ) ⩽ 2 L y (τ) for a.e. τ \in (0, t), y (0) = ‖ u_{0} ‖_{2, Ω}^{2} . \end{matrix}$ Dropping the second term on the left-hand side and integrating we obtain the estimate: for all $τ \in [0, t]$ $\begin{matrix} (5.14) & y (τ) = {‖ u (τ) ‖}_{2, Ω}^{2} ⩽ ‖ u_{0} ‖_{2, Ω}^{2} e^{2 L τ} ⩽ ‖ u_{0} ‖_{2, Ω}^{2} e^{2 L t} . \end{matrix}$ Thus, we can choose $M (t) : = ‖ u_{0} ‖_{2, Ω}^{2} e^{2 L t}$ . Plugging this expression for $M (t)$ into (5.13) and using (5.11) (b) we come to the inequality $\begin{matrix} y^{'} (t) + 2 κ e^{2 L (p^{-} - p^{+}) t} y^{\frac{p^{+}}{2}} (t) ⩽ 2 L y (t) for a.e. in (0, T), y (0) = ‖ u_{0} ‖_{2, Ω}^{2}, \end{matrix}$ which is studied in exactly the same way as inequality (5.5). □

Theorem 5.5.

Let $u (z)$ be a nontrivial strong solution of problem ( 1.1 ). Assume that the exponents p, q and the coefficients D, A satisfy conditions (H1), (H2) and that the following conditions are fulfilled: $\begin{matrix} (5.15) & \begin{matrix} (a) & p^{-} > s > \frac{2 n}{n + 2}, q^{-} > 1, A^{\frac{- 1}{q (x) - 1}} (x) \in L^{1} (Ω), \frac{1}{σ^{+}} > 1, \\ condition (H3) holds with f_{0} = 0 and L > 0 . \\ (b) & ‖ u_{0} ‖_{2, Ω}^{2 (1 - σ^{+})} < min {1, \frac{γ}{L} \frac{1 - σ^{+}}{1 - σ^{-}}} with the constants γ, σ^{\pm} defined in (5.7) . \end{matrix} \end{matrix}$ Then $u (z)$ vanishes at the finite moment: $\begin{matrix} {‖ u (t) ‖}_{2, Ω} = 0 for all t ⩾ t^{*} = \frac{- 1}{2 L (1 - σ^{-})} ln (1 - \frac{L}{γ} \frac{1 - σ^{-}}{1 - σ^{+}} ‖ u_{0} ‖_{2, Ω}^{2 (1 - σ^{+})}) . \end{matrix}$

Proof.

We repeat the proof of Theorem 5.2 substituting the constant M by the function $M (t)$ introduced in (5.12). Combining (5.9), (5.14), and taking into account (5.15) we arrive at the differential inequality for $y (t)$ , $\begin{matrix} y^{'} (t) + 2 γ e^{2 L (σ^{-} - σ^{+}) t} y^{σ^{+}} (t) ⩽ 2 L y (t) for a.e. t \in (0, T), y (0) = ‖ u_{0} ‖_{2, Ω}^{2}, \end{matrix}$ and the conclusion follows in the standard way. □

Footnotes

Acknowledgements

The authors would like to thank the anonymous referee for valuable remarks and suggestions.

The work was partially supported by CSF-CAPES-PVE – Process 88881.030388/2013-01, Brazil. The first author acknowledges the support of the Research Grants MTM2013-43671-P, MTM2017-87162-P, MICINN, Spain, and the program “Science Without Borders”, CSF-CAPES-PVE-Process 88887.059583/2014-00, Brazil. The second author was supported with CNPq scholarship – process 202645/2014-2, Brazil. The third author was supported with CAPES scholarship – process 99999.006893/2014-07, Brazil.

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Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

Abstract

Keywords

1. Introduction

1.1. Statement of the problem

1.2. Previous work

1.3. Organization of the paper and results

2. Weighted spaces with variable exponents

2.1. Lebesgue and Sobolev spaces with variable exponents

2.2. Weighted spaces

Proposition 2.1 (The dual space of L B r ( Ω ) ).

Proposition 2.3 (Theorem 8.3.1, [9]).

Proposition 2.6 (Lemmas 2.5, 2.6, 2.7, [2]).

3. Existence and uniqueness of weak solutions

Footnotes

Acknowledgements

References

Proposition 2.1 (The dual space of $L_{B}^{r} (Ω)$ ).