1 Introduction

In this paper, we consider the following semilinear strongly degenerate parabolic equation

$$ \left\{ \begin{array}{ll} u_{t} -{\Delta}_{\lambda} u + f(u) = g(x),\quad & x\in{\Omega}, t>0,\\ u(x,t)=0, & x\in\partial {\Omega}, t>0,\\ u(x,0)=u_{0}(x), & x\in{\Omega}, \end{array}\right. $$
(1)

where Ω is a bounded domain in \(\phantom {\dot {i}\!}\mathbb {R}^{N} (N\ge 2)\) with smooth boundary ∂Ω, u 0,g ∈ L 2(Ω), the nonlinear term f(u) satisfies certain conditions specified later, and Δ λ is a strongly degenerate operator of the form

$${\Delta}_{\lambda} := \sum\limits_{i = 1}^{N} \partial_{x_{i}}({\lambda_{i}^{2}}(x)\partial_{x_{i}}), $$

where \(\phantom {\dot {i}\!}\lambda = (\lambda _{1},\dots ,\lambda _{N}):\mathbb {R}^{N} \to \mathbb {R}^{N}\) satisfies certain conditions specified below. This operator was introduced by Franchi and Lanconelli in [9] (see also [8]) and recently reconsidered in [12] under an additional assumption that the operator is homogeneous of degree two with respect to a group dilation in \(\phantom {\dot {i}\!}\mathbb R^{N}\). Here, the functions \(\phantom {\dot {i}\!}\lambda _{i} :\mathbb {R}^{N} \to \mathbb {R}\) are continuous, strictly positive and of class C 1 outside the coordinate hyperplanes, i.e., λ i > 0, i = 1,…,N in \(\phantom {\dot {i}\!}\mathbb {R}^{N}\setminus \prod , \)where \(\phantom {\dot {i}\!}\prod =\{(x_{1}, \ldots , x_{N})\in \mathbb {R}^{N}: {\prod }^{N}_{i=1}x_{i} =0\}\). As in [12], we assume that λ i satisfy the following properties:

  1. 1.

    λ 1(x)≡1, λ i (x) = λ i (x 1,…,x i−1), i = 2,…,N;

  2. 2.

    For every \(\phantom {\dot {i}\!}x\in \mathbb {R}^{N}\), λ i (x) = λ i (x ∗),i = 1,…,N, where

    $$x^{\ast}= (|x_{1}|, \ldots, |x_{N}|) \text{ if } x = (x_{1}, \ldots, x_{N}); $$
  3. 3.

    There exists a constant ρ ≥ 0 such that

    $$0 \le x_{k}\partial_{x_{k}}\lambda_{i}(x) \le \rho \lambda_{i}(x)\quad \forall k \in \{1, \ldots, i-1\}, i =2, \ldots, N, $$

    and for every \(\phantom {\dot {i}\!}x\in \mathbb {R}^{N}_{+}:=\{(x_{1}, \ldots , x_{N})\in \mathbb {R}^{N}: x_{i} \ge 0 \forall i=1, \ldots , N\}\);

  4. 4.

    There exists a group of dilations {δ t } t>0

    $$ \delta_{t} :\mathbb{R}^{N} \to \mathbb{R}^{N},\quad \delta_{t}(x) =\delta_{t}(x_{1}, \ldots, x_{N}) =(t^{\epsilon_{1}}x_{1}, \ldots, t^{\epsilon_{N}}x_{N}), $$

    where 1 ≤ 𝜖 1 ≤ 𝜖 2 ≤ ⋯ ≤ 𝜖 N such that λ i is δ t -homogeneous of degree 𝜖 i −1,i.e.,

    $$ \lambda_{i}(\delta_{t}(x)) =t^{\epsilon_{i} -1}\lambda_{i}(x) \quad \forall x\in \mathbb{R}^{N}, t>0, i=1, \ldots, N. $$

    This implies that the operator Δ λ is δ t -homogeneous of degree two, i.e.,

    $${\Delta}_{\lambda}(u(\delta_{t}(x))) =t^{2}({\Delta}_{\lambda} u)(\delta_{t}(x)) \quad \forall u \in C^{\infty}(\mathbb{R}^{N}). $$

We denote by Q the homogeneous dimension of \(\phantom {\dot {i}\!}\mathbb {R}^{N}\) with respect to the group of dilations {δ t } t>0,i.e.,

$$Q:=\epsilon_{1}+\cdots+\epsilon_{N}.</p><p class="noindent">$$

The homogeneous dimension Q plays a crucial role, both in the geometry and the functional associated to the operator Δ λ .

The Δ λ -Laplace operator contains many degenerate elliptic operators such as the Grushin type operator

$$ G_{\alpha} ={\Delta}_{x}+|x|^{2\alpha}{\Delta}_{y}, \quad \alpha>0, $$

where (x,y) denotes the point of \(\phantom {\dot {i}\!}\mathbb {R}^{N_{1}}\times \mathbb {R}^{N_{2}}\), and the strongly degenerate operator of the form

$$P_{\alpha, \beta} = {\Delta}_{x} + {\Delta}_{y} + |x|^{2\alpha}|y|^{2\beta} {\Delta}_{z} , $$

where \(\phantom {\dot {i}\!}(x, y, z) \in \mathbb {R}^{N_{1}}\times \mathbb {R}^{N_{2}}\times \mathbb {R}^{N_{3}}\) ( N i ≥ 1, i = 1, 2, 3), α, β are real positive constants, see [19]. We refer the interested reader to [13, Section 2.3] for other examples of Δ λ -Laplacians. See also [4, 16] for recent results related to elliptic equations involving this operator.

In the last years, the existence and long-time behavior in terms of existence of global attractors of solutions to semilinear parabolic equations involving the above degenerate operators have been studied extensively by a number of authors. Up to now, there are two main kinds of nonlinearities that have been considered. The first one is the class of nonlinearities that is locally Lipschitzian continuous and satisfies a Sobolev growth condition

$$|f(u) - f(v)| \le C(1 + |u|^{\rho} + |v|^{\rho})|u - v|,\quad 0 \leq \rho < \frac{4}{Q- 2}, $$

and some suitable dissipative conditions; see [2, 13–15, 20]. The second one is the class of nonlinearities that satisfies a polynomial growth

$$\begin{array}{@{}rcl@{}} C_{1}|u|^{p} - C_{0} \le f(u)u &\le& C_{2}|u|^{p} + C_{0}\quad \text{ for some } p \ge 2,\\ f^{\prime}(u) &\ge& - \ell, \end{array} $$

see [3, 6, 18, 20]. See also some related results in the case of unbounded domains [1, 5], the more delicated case due to the lack of compactness of the Sobolev type embeddings.

Note that for both above classes of nonlinearities, some restriction on the upper growth of the nonlinearity is imposed and an exponential nonlinearity, for example, f(u) = e u, does not hold. In this paper, we try to remove this restriction and we were able to prove the existence of weak solutions and existence of global attractors for a very large class of nonlinearities that particularly covers both above classes and even exponential nonlinearities. This is the main novelty of our paper.

To study problem (1), we assume that the initial datum u 0 ∈ L 2(Ω) is given, the nonlinearity f and the external force g satisfy the following conditions:

  • (F) \(\phantom {\dot {i}\!}f: \mathbb {R} \to \mathbb {R}\) is a continuously differentiable function satisfying

    $$\begin{array}{@{}rcl@{}} f^{\prime}(u) &\ge& -\ell, \end{array} $$
    (2)
    $$\begin{array}{@{}rcl@{}} f(u)u &\ge& - \mu u^{2} - {C_{1}}, \end{array} $$
    (3)

    where C 1, ℓ are two positive constants, 0 < μ < γ 1 with γ 1 > 0 is the first eigenvalue of the operator −Δ λ in Ω with the homogeneous Dirichlet boundary condition, and \(F(u) = {{\int }_{0}^{u}} f(s)ds\) is a primitive of f;

  • (G) g ∈ L 2(Ω).

It follows from (2) that \(\phantom {\dot {i}\!}0\leq {{\int }_{0}^{u}} (f^{\prime }(s)s+\ell s)ds\), and therefore by integrating by parts, we obtain

$$ F(u) \leq f(u)u+ \ell \frac{u^{2}}{2}\quad \text{ for all } u\in \mathbb{R}. $$
(4)

To study problem (1), we use the weighted Sobolev space \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })\) defined as the completion of \(\phantom {\dot {i}\!}{C^{1}_{0}}({\Omega })\) in the norm

$$\|u\|_{{\overset{\circ}W}{^{1,2}_{\lambda}}({\Omega})}:=\left( {\int}_{\Omega} |\nabla_{\lambda}u|^{2}dx\right)^{1/2}. $$

This is a Hilbert space with respect to the following scalar product

$$((u,v))_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}={\int}_{\Omega} \nabla_{\lambda}u \cdot \nabla_{\lambda}vdx. $$

We also use the Hilbert space D(Δ λ ) defined as the domain of the operator −Δ λ with the homogeneous Dirichlet boundary condition

$$D({\Delta}_{\lambda})=\left\{u\in {\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})} | {\Delta}_{\lambda} u\in L^{2}({\Omega})\right\}, $$

with the graph norm

$$\|u\|_{D({\Delta}_{\lambda})}:=\left( {\int}_{\Omega} |{\Delta}_{\lambda}u|^{2}dx\right)^{1/2}. $$

By the result in [12], we know that the embedding \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })\hookrightarrow L^{2}({\Omega })\) is compact. Using this embedding and the definition of D(Δ λ ), we will show in Lemma 4 below that the embedding \(\phantom {\dot {i}\!}D({\Delta }_{\lambda }) \hookrightarrow {\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })\) is also compact. These compact embeddings will play an important role for our investigation.

Let γ 1 > 0 be the first eigenvalue of the operator −Δ λ in Ω with homogeneous Dirichlet boundary conditions. Then

$$\gamma_{1}=\inf \left\{\frac{\|u\|^{2}_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}}{\|u\|^{2}_{L^{2}({\Omega})}}: u \in {\overset{\circ}{W}}{^{1, 2}_{\lambda}}({\Omega})\setminus \{0\}\right\}. $$

Therefore,

$$ \|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} \ge \gamma_{1}\|u\|_{L^{2}({\Omega})}^{2} \quad\text{ for all } u \in {\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})}. $$
(5)

The paper is organized as follows. In Section 2, we prove the existence and uniqueness of weak solutions by utilizing the compactness method and weak convergence techniques in Orlicz spaces [11]. In Section 3, we prove the existence of global attractors for the semigroup generated by the problem in various spaces. The main novelty of the paper is that the nonlinearity can grow exponentially.

2 Existence and Uniqueness of Weak Solutions

Definition 1

A function u is called a weak solution of problem (1) on (0,T) if \(\phantom {\dot {i}\!}u \in C([0,T];{L^{2}}({\Omega })) \cap {L^{2}}(0,T;{\overset {\circ }{W}}{^{1,2}_{\lambda }}{({\Omega }))}\), f(u) ∈ L 1(Q T ), u(0) = u 0, \(\phantom {\dot {i}\!}\frac {du}{dt}\in \) \(\phantom {\dot {i}\!}L^{2}(0,T;({\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }))^{\ast })+L^{1}(Q_{T})\), and

$$\frac{du}{dt} - {\Delta}_{\lambda} u + f(u) = g \quad\text{ in } L^{2}(0,T;({\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega}))^{\ast})+L^{1}(Q_{T}) $$

or equivalently,

$$\left\langle \frac{du}{dt} - {\Delta}_{\lambda} u + f(u), w\right\rangle = \langle g, w\rangle $$

for all test functions \(\phantom {\dot {i}\!}w \in W:={\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }) \cap L^{\infty }({\Omega })\) and for a.e. t∈(0,T). Here, 〈⋅,⋅〉 denotes the dual bracket between W and its dual W ∗, and \(\phantom {\dot {i}\!}({\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }))^{\ast }\) is the dual space of \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })\).

Theorem 1

Assume (F)–(G) hold. Then for any u 0 ∈ L 2 (Ω) and T > 0 given, problem (1) has a unique weak solution u on the interval (0,T). Moreover, the mapping u 0 ↦ u(t) is continuous on L 2 (Ω), that is, the solutions depend continuously on the initial data.

Proof

i) Existence. We will prove the existence of a weak solution by using the compactness method. The main difference compared with the proofs in [3, 20] is that the nonlinear term f(u) only belongs to L 1(Q T ) due to no restriction imposed on its upper growth. This introduces some essential difficulties when establishing a priori estimates and passing to the limit for the nonlinear term.

Let {u n } be the Galerkin appropriate solutions. We will establish some a priori estimates for u n . We have

$$ \frac{1}{2}\frac{d}{dt}\|u_{n}\|_{L^{2}({\Omega})}^{2} + \|u_{n}\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} f(u_{n})u_{n}dx = {\int}_{\Omega} gu_{n}dx. $$
(6)

Therefore,

$$\begin{array}{@{}rcl@{}} &&\frac{1}{2}\frac{d}{dt}\|u_{n}(t)\|_{L^{2}({\Omega})}^{2} + \|u_{n}(t)\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} - \mu\|u_{n}(t)\|_{L^{2}({\Omega})}^{2} - C_{1}|{\Omega}|\\ &&\quad\le \frac{1}{2\varepsilon}\|g\|_{L^{2}({\Omega})}^{2} + \frac{\varepsilon}{2}\|u_{n}(t)\|_{L^{2}({\Omega})}^{2}. \end{array} $$

Using the inequality (5), we get

$$\begin{array}{@{}rcl@{}} &&\frac{d}{dt}\|u_{n}(t)\|_{L^{2}({\Omega})}^{2} + \varepsilon\|u_{n}(t)\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + (2\gamma_{1} - 2\mu - \varepsilon \gamma_{1} - \varepsilon)\|u_{n}(t)\|_{L^{2}({\Omega})}^{2}\\ &&\quad \le\frac{1}{\varepsilon}\|g\|_{L^{2}({\Omega})}^{2} +2C_{1}|{\Omega}|, \end{array} $$

where ε > 0 is small enough so that 2γ 1−2μ−ε γ 1−ε > 0. Integrating from 0 to t, 0 ≤ t ≤ T, we get

$$\begin{array}{@{}rcl@{}} &&\|u_{n}(t)\|_{L^{2}({\Omega})}^{2} + \varepsilon {{\int}_{0}^{t}} \|u_{n}(s)\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2}ds + (2\gamma_{1} - 2\mu - \varepsilon \gamma_{1} - \varepsilon){{\int}_{0}^{t}} \|u_{n}(s)\|_{L^{2}({\Omega})}^{2}ds\\ &&\quad\le \frac{1}{\varepsilon}\|g\|_{L^{2}({\Omega})}^{2}T + 2C_{1}|{\Omega}|T + \|u(0)\|_{L^{2}({\Omega})}^{2}. \end{array} $$

This inequality yields

$$\begin{array}{@{}rcl@{}} &&\{u_{n}\} \text{ is bounded in } L^{\infty}(0,T;L^{2}({\Omega})),\\ &&\{u_{n}\} \text{ is bounded in } L^{2}(0,T;{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})). \end{array}$$

Due to the boundedness of {u n } in \(\phantom {\dot {i}\!}L^{2}(0,T;{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }))\), it is easy to check that {Δ λ u n } is bounded in \(\phantom {\dot {i}\!}L^{2}(0,T;({\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }))^{\ast })\). From the above results, we can assume that (up to a subsequence)

$$\begin{array}{@{}rcl@{}} u_{n} &\rightharpoonup& u \text{ in } L^{2}(0,T;{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})),\\ u_{n} &\rightharpoonup^{\ast}& u \text{ in } L^{\infty}(0,T;L^{2}({\Omega})),\\ {\Delta}_{\lambda} u_{n} &\rightharpoonup& {\Delta}_{\lambda} u \text{ in } L^{2}(0,T;({\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega}))^{\ast}). \end{array} $$

On the other hand, using the Cauchy inequality in (6), we have

$$\frac{1}{2}\frac{d}{dt}\|u_{n}\|_{L^{2}({\Omega})}^{2} + \|u_{n}\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} f(u_{n})u_{n}dx \leq \frac{1}{2\gamma_{1}} \|g\|^{2}_{L^{2}({\Omega})}+\frac{\gamma_{1}}{2}\|u_{n}\|^{2}_{L^{2}({\Omega})}. $$

Noting that \(\phantom {\dot {i}\!}\|u_{n}\|_{{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })}^{2}\geq \gamma _{1}\|u_{n}\|^{2}_{L^{2}({\Omega })}\), integrating this inequality from 0 to T, we have

$${{\int}_{0}^{T}} \|u_{n}\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2}ds + 2 {\int}_{Q_{T}} f(u_{n})u_{n}dxdt \le \|u_{0}\|_{L^{2}({\Omega})}^{2} + \frac{1}{\gamma_{1}}\|g\|_{L^{2}({\Omega})}^{2}T. $$

Hence,

$$ {\int}_{Q_{T}} f(u_{n})u_{n}dxdt \le C. $$
(7)

We now prove that {f(u n )} is bounded in L 1(Q T ). Putting h(s) = f(s)−f(0) + γ s, where γ > ℓ and noting that h(s)s = (f(s)−f(0))s + γ s 2 = f ′(c)s 2 + γ s 2 ≥ (γ−ℓ)s 2 ≥ 0 for all \(\phantom {\dot {i}\!}s\in \mathbb {R}\), we have

$$\begin{array}{@{}rcl@{}} {\int}_{Q_{T}} |h(u_{n})|dxdt &\le& {\int}_{Q_{T} \cap \{|u_{n}| > 1\}}|h(u_{n})u_{n}|dxdt + {\int}_{Q_{T} \cap \{|u_{n}|\le 1\}} |h(u_{n})|dxdt\\ &\le& {\int}_{Q_{T}} h(u_{n})u_{n}dxdt +\sup_{|s| \le 1} |h(s)| |Q_{T}|\\ &\le& {\int}_{Q_{T}} f(u_{n})u_{n}dxdt + |f(0)|\|u_{n}\|_{L^{1}(Q_{T})}+\gamma \|u_{n}\|^{2}_{L^{2}(Q_{T})}\\ &&+ \sup_{|s| \le 1}|h(s)| |Q_{T}|\\ &\le& C, \end{array} $$

where we have used (7) and the boundedness of {u n } in L ∞(0,T;L 2(Ω)). Hence, it implies that {h(u n )}, and therefore, {f(u n )} is bounded in L 1(Q T ). Since

$$\frac{du_{n}}{dt} = {\Delta}_{\lambda} u_{n} - f(u_{n}) + g, $$

we deduce that \(\phantom {\dot {i}\!}\{\frac {du_{n}}{dt}\}\) is bounded in \(\phantom {\dot {i}\!}L^{2}(0,T;({\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }))^{\ast })+L^{1}(Q_{T})\), and therefore in \(\phantom {\dot {i}\!}L^{1}(0,T;({\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }))^{\ast }+L^{1}({\Omega }))\). Because \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }) \subset \subset L^{2}({\Omega }) \subset ({\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }))^{\ast }+L^{1}({\Omega })\), by the Aubin–Lions–Simon compactness lemma (see [7]), we have that {u n } is compact in L 2(0,T;L 2(Ω)). Hence, we may assume, up to a subsequence, that u n →u a.e. in Q T . Applying Lemma 6.1 in [10], we obtain that h(u) ∈ L 1(Q T ) and for all test function \(\phantom {\dot {i}\!}\xi \in C_{0}^{\infty } ([0,T];{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }) \cap L^{\infty }({\Omega }))\),

$${\int}_{Q_{T}} h(u_{n})\xi dxdt \to {\int}_{Q_{T}} h(u)\xi dxdt. $$

Hence, f(u) ∈ L 1(Q T ) and

$${\int}_{Q_{T}} f(u_{n})\xi dxdt \to {\int}_{Q_{T}} f(u)\xi dxdt\quad \text{for all } \xi \in C_{0}^{\infty} ([0,T];{\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})} \cap L^{\infty}({\Omega})).$$

Thus, u satisfies (6). Repeating the arguments in [3], we get u(0) = u 0 and this implies that u is a weak solution to problem (1).

ii) Uniqueness and continuous dependence on the initial data. Let u and v be two weak solutions of (1) with initial data u 0,v 0 ∈ L 2(Ω). Putting w = u−v, we have

$$ \left\{\begin{array}{l} w_{t} -{\Delta}_{\lambda} w + \widetilde f(u) - \widetilde f(v) -\ell w=0,\\ w(0)=u_{0}-v_{0}, \end{array}\right. $$
(8)

where \(\phantom {\dot {i}\!}\widetilde {f}(s) = f(s) + \ell s\). Here, because w(t) does not belong to \(\phantom {\dot {i}\!}W: = {\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega }) \cap {L^{\infty }}({\Omega })\), we cannot choose w(t) as a test function as in [3]. Consequently, the proof will be more involved.

We use some ideas in [11]. Let \(\phantom {\dot {i}\!}B_{k}: \mathbb {R} \to \mathbb {R}\) be the truncated function

$$B_{k}(s) = \left\{\begin{array}{ll} k &\text{ if } s > k,\\ s &\text{ if } |s| \le k,\\ - k&\text{ if } s < - k. \end{array}\right. $$

Consider the corresponding Nemytskii mapping \(\phantom {\dot {i}\!}\widehat {B}_{k}:W \to W\) defined as follows

$$\widehat{B}_{k}(w)(x) = B_{k}(w(x))\quad \text { for all } x\in {\Omega}. $$

By Lemma 2.3 in [11], we have that \(\phantom {\dot {i}\!}\|\widehat {B}_{k}(w) - w\|_{W} \to 0\) as k→∞. Now multiplying the first equation in (8) by \(\phantom {\dot {i}\!}\widehat {B}_{k}(w)\), then integrating over Ω×(ε,t), where t∈(0,T), we get

$$\begin{array}{@{}rcl@{}} &&{\int}_{\varepsilon}^{t} {\int}_{\Omega} \frac{d}{ds}\left( w(s) \widehat {B}_{k}(w)(s)\right)dxds - {\int}_{\varepsilon}^{t} {\int}_{\Omega} w \frac{d}{ds}\left( \widehat{B}_{k}(w)(s)\right)dxds\\ && + \frac{1}{2}{\int}_{\varepsilon}^{t} {\int}_{\{ x \in {\Omega}:|w(x,s)| \le k\}}|\nabla_{\lambda} w|^{2}dxds\\ && +{\int}_{\varepsilon}^{t} {\int}_{\Omega} (\widetilde f(u) - \widetilde f(v))\widehat{B}_{k}(w)dxds -\ell {\int}_{\varepsilon}^{t} {\int}_{\Omega} w{\widehat B}_{k}(w)dxds=0. \end{array} $$

Noting that \(\phantom {\dot {i}\!}w\frac {d}{dt}(\widehat {B}_{k}(w))=\frac {1}{2}\frac {d}{dt}((\widehat {B}_{k}(w))^{2})\), we have

$$\begin{array}{@{}rcl@{}} &&{\int}_{\Omega} w(t)\widehat{B}_{k}(w)(t)dx - \frac{1}{2}\left\|\widehat{B}_{k}(w)(t)\right\|_{L^{2}({\Omega})}^{2} + \frac{1}{2}{\int}_{\varepsilon}^{t} {\int}_{\{x \in {\Omega}:|w(x,s)| \le k\}} |\nabla_{\lambda} w|^{2}dxds\\ && +{\int}_{\varepsilon}^{t} {\int}_{\Omega} \widetilde{f^{\prime}}(\xi) w\widehat{B}_{k}(w)dxds\\ &&= {\int}_{\Omega} w(\varepsilon)\widehat{B}_{k}(w)(\varepsilon)dx - \frac{1}{2}\left\|\widehat{B}_{k}(w)(\varepsilon)\right\|_{L^{2}({\Omega})}^{2} + \ell{\int}_{\varepsilon}^{t} {\int}_{\Omega} w\widehat{B}_{k}(w)dxds. \end{array} $$

Note that \(\phantom {\dot {i}\!}\widetilde {f}^{\prime }(s) \ge 0\) and s B k (s) ≥ 0 for all \(\phantom {\dot {i}\!}s\in \mathbb {R}\), by letting ε→0 and k→∞ in the above equality, we obtain

$$\|w(t)\|_{L^{2}({\Omega})}^{2} \le \|w(0)\|_{L^{2}({\Omega})}^{2} + 2\ell {{\int}_{0}^{t}} \|w(s)\|_{L^{2}({\Omega})}^{2}ds. $$

Hence, by the Gronwall inequality of integral form, we get

$$\|w(t)\|_{L^{2}({\Omega})}^{2} \le \|w(0)\|_{L^{2}({\Omega})}^{2} e^{2\ell t} \leq \|w(0)\|_{L^{2}({\Omega})}^{2} e^{2\ell T}\quad \text{ for all } t \in [0,T]. $$

Note that w ∈ C([0,T];L 2(Ω)), in particular, we get the uniqueness if w(0) = 0.

3 Existence of a Global Attractor

By Theorem 1, we can define a continuous (nonlinear) semigroup S(t):L 2(Ω)→L 2(Ω) associated to problem (1) as follows

$$S(t)u_{0}:=u(t), $$

where u(⋅) is the unique weak solution of (1) with the initial datum u 0. We will prove that the semigroup S(t) has a global attractor \(\phantom {\dot {i}\!}\mathcal {A}\) in the space \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })\).

For brevity, in the following lemmas, we give some formal calculation, the rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [17].

Lemma 1

The semigroup {S(t)} t ≥ 0 has a bounded absorbing set in L 2 (Ω).

Proof

Multiplying the first equation in (1) by u, we have

$$ \frac{1}{2}\frac{d}{dt}\|u\|_{L^{2}({\Omega})}^{2} + \|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} f(u)udx = {\int}_{\Omega} gudx. $$
(9)

Using inequalities (3), (5), and the Cauchy inequality, we arrive at

$$\frac{d}{dt}\|u\|_{L^{2}({\Omega})}^{2} + (\gamma_{1} - \mu)\|u\|_{L^{2}({\Omega})}^{2} \le 2C_{1}|{\Omega}| + \frac{1}{\gamma_{1} - \mu}\|g\|_{L^{2}({\Omega})}^{2}. $$

Hence, thanks to the Gronwall inequality, we obtain

$$\|u\|_{L^{2}({\Omega})}^{2} \le \|u_{0}\|_{L^{2}({\Omega})}^{2}e^{-(\gamma_{1} - \mu)t} + R_{1}, $$

where \(\phantom {\dot {i}\!}R_{1} = R_{1}(\gamma _{1},\mu ,|{\Omega }|,\|g\|_{L^{2}({\Omega })})\). Hence, choosing ρ 1 = 2R 1, we have

$$ \|u\|_{L^{2}({\Omega})}^{2} \le \rho_{1}\quad \text{ for all } t \ge T_{1} = T_{1}(\|u_{0}\|_{L^{2}({\Omega})}). $$
(10)

This completes the proof.

Lemma 2

The semigroup {S(t)} t ≥ 0 has a bounded absorbing set in \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })\).

Proof

Multiplying the first equation in (1) by −Δ λ u and integrating by parts, we obtain

$$\begin{array}{@{}rcl@{}} \frac{1}{2}\frac{d}{dt}\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + \|{\Delta}_{\lambda} u\|_{L^{2}({\Omega})}^{2}&=& - {\int}_{\Omega} f^{\prime}(u)|\nabla_{\lambda} u|^{2}dx - {\int}_{\Omega} g{\Delta}_{\lambda} udx\\ &\le& \ell\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + \frac{1}{2}\|g\|_{L^{2}({\Omega})}^{2} + \frac{1}{2}\|{\Delta}_{\lambda} u\|_{L^{2}({\Omega})}^{2}. \end{array} $$

In particular,

$$ \frac{d}{dt}\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} \le 2\ell\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + \|g\|_{L^{2}({\Omega})}^{2}. $$
(11)

On the other hand, integrating (9) from t to t+1 and using (3), we have

$$\begin{array}{@{}rcl@{}} &&{\int}_{t}^{t + 1} \|u(s)\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2}ds + \frac{1}{2} \|u(t + 1)\|_{L^{2}({\Omega})}^{2}\\ &&\le (\mu+1) {\int}_{t}^{t + 1} \|u(s)\|_{L^{2}({\Omega})}^{2}ds + \frac{1}{2}\|u(t)\|_{L^{2}({\Omega})}^{2} + C_{1}|{\Omega}| + \frac{1}{4}\|g\|_{L^{2}({\Omega})}^{2}\\ &&\le \rho_{2} = \rho_{2}(\gamma_{1},\mu,|{\Omega}|,\|g\|_{L^{2}({\Omega})}) \end{array} $$
(12)

for all \(\phantom {\dot {i}\!}t \ge {T_{1}}={T_{1}}(\|u_{0}\|_{L^{2}({\Omega })})\), where we have used the Cauchy inequality and estimate (10). By the uniform Gronwall inequality, from (11) and (12), we deduce that

$$ \|u(t)\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} \le \rho_{2} \quad \text{ for all } t \ge {T_{2}} = {T_{1}}+1. $$
(13)

This completes the proof.

Lemma 3

The semigroup {S(t)} t ≥ 0 has a bounded absorbing set in D(Δ λ ).

Proof

By differentiating (1) in time, we get

$$u_{tt} - {\Delta}_{\lambda} u_{t} + f^{\prime}(u)u_{t} = 0. $$

Taking the inner product of this equality with u t in L 2(Ω) and using (2), in particular, we obtain

$$ \frac{1}{2}\frac{d}{dt}\|u_{t}\|_{L^{2}({\Omega})}^{2} \le \ell \|u_{t}\|_{L^{2}({\Omega} )}^{2}. $$
(14)

Multiplying the first equation in (1) by u t , we obtain

$$ \frac{d}{dt}\left( \frac{1}{2}\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} F(u)dx - {\int}_{\Omega} gudx \right) = -\|u_{t}\|_{L^{2}({\Omega})}^{2} \le 0. $$
(15)

On the other hand, integrating (9) from t to t+1 and using (10), we have

$${\int}_{t}^{t + 1} \left[\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} f(u)udx - {\int}_{\Omega} gudx\right] ds \le \|u(t)\|_{L^{2}({\Omega})}^{2} \le \rho_{1} \quad \forall t \ge T_{1}. $$

Using the inequality (4), we deduce that

$$\begin{array}{@{}rcl@{}} &&{\int}_{t}^{t + 1} \left[\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} f(u)udx - {\int}_{\Omega} gudx \right] ds\\ &&\ge {\int}_{t}^{t + 1} \left[\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} F(u)dx - \frac{\ell}{2}\|u\|_{L^{2}({\Omega})}^{2} - {\int}_{\Omega} gudx\right]ds\\ &&\ge {\int}_{t}^{t + 1} \left[\frac{1}{2}\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} F(u)dx - {\int}_{\Omega} gudx\right]ds -\frac{\ell}{2}\rho_{1}\quad \text{ for all } t\geq T_{1}, \end{array} $$

where we have used the inequality (10). Hence,

$$ {\int}_{t}^{t + 1} \left[\frac{1}{2}\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} F(u)dx - {\int}_{\Omega} gudx \right] ds \le \left( 1+\frac{\ell}{2}\right)\rho_{1}\quad \forall t \ge T_{1}. $$
(16)

By the uniform Gronwall inequality, from (15) and (16), we deduce that

$$\frac{1}{2}\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + {\int}_{\Omega} F(u)dx - {\int}_{\Omega} gudx \le \rho_{3}\quad \text {for all } t \ge T_{2} = T_{1} + 1. $$
(17)

Integrating (15) from t to t+1 and using (17), we obtain

$$ {\int}_{t}^{t + 1} \|u_{t}\|_{L^{2}({\Omega})}^{2}ds \le \rho_{3}\quad \text { for all } t \ge T_{2}. $$
(18)

Combining (14) with (18) and using the uniform Gronwall inequality, we have

$$ \|u_{t}\|_{L^{2}({\Omega})}^{2} \le \rho_{3}\quad \text { for all } t \ge T_{3} =T_{2}+1. $$
(19)

On the other hand, multiplying the first equation in (1) by −Δ λ u, using (2) and the Cauchy inequality, we obtain

$$\begin{array}{@{}rcl@{}} \|{\Delta}_{\lambda} u\|_{L^{2}({\Omega})}^{2}&=& {\int}_{\Omega} u_{t}{\Delta}_{\lambda} udx - {\int}_{\Omega} f^{\prime}(u)|\nabla_{\lambda} u|^{2}dx - {\int}_{\Omega} g{\Delta}_{\lambda} udx\\ &\le& \ell\|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}^{2} + \|u_{t}\|_{L^{2}({\Omega})}^{2} + \frac{1}{2}\|{\Delta}_{\lambda} u\|_{L^{2}({\Omega})}^{2} + \|g\|_{L^{2}({\Omega})}^{2}. \end{array} $$

Using estimates (13) and (19), we arrive at

$$\|{\Delta}_{\lambda} u(t)\|_{L^{2}({\Omega})}^{2} \le \rho_{4}\quad \text { for all } t \ge T_{3}. $$

This completes the proof.

We now prove the following important lemma.

Lemma 4

The embedding \(\phantom {\dot {i}\!}D({\Delta }_{\lambda }) \hookrightarrow {\overset {\circ }{W}}{^{1,2}_{\lambda }}({\Omega })\) is compact.

Proof

First, for any u ∈ D(Δ λ ), we have

$$ \|u\|^{2}_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}({\Omega})}={\int}_{\Omega} |\nabla_{\lambda} u|^{2}dx=-{\int}_{\Omega} u\cdot{\Delta}_{\lambda} u dx\leq \|u\|_{L^{2}({\Omega})}\|{\Delta}_{\lambda} u\|_{L^{2}({\Omega})}. $$
(20)

Next, we will prove that for any ε > 0, there exists C(ε) such that

$$ \|u\|^{2}_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})}}\leq \varepsilon \|{\Delta}_{\lambda} u\|^{2}_{L^{2}({\Omega})}+C(\varepsilon)\|u\|^{2}_{L^{1}({\Omega})} $$
(21)

for all u ∈ D(Δ λ ). Indeed, since \(\phantom {\dot {i}\!}\overset {\circ }{W}{^{1,2}_{\lambda }}({\Omega })\subset \subset L^{2}({\Omega })\subset L^{1}({\Omega })\), by the Ehrling lemma, we have for any η > 0,

$$\|u\|_{L^{2}({\Omega})}\leq \eta \|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})}}+C_{1}(\eta)\|u\|_{L^{1}({\Omega})}. $$

Substituting this inequality into (20), we obtain

$$\begin{array}{@{}rcl@{}} \|u\|^{2}_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})}}&\leq& \|{\Delta}_{\lambda} u\|_{L^{2}({\Omega})}\left( \eta \|u\|_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})}}+C_{1}(\eta)\|u\|_{L^{1}({\Omega})}\right)\\ &\leq& \eta \|u\|^{2}_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})}} + \eta \|{\Delta}_{\lambda} u\|^{2}_{L^{2}({\Omega})}+C_{2}(\eta)\|u\|^{2}_{L^{1}({\Omega})}, \end{array} $$

where we have used the Cauchy inequality. Hence, we obtain (21) for suitable choosing of η.

We now prove the compactness of the embedding \(D({\Delta }_{\lambda }) \hookrightarrow {\overset {\circ }{W}}{^{1,2}_{\lambda }}{({\Omega })}\). Let {u n } be a bounded sequence in D(Δ λ ). Then there exists a subsequence \(\phantom {\dot {i}\!}\{u_{n_{k}}\}\) such that \(\phantom {\dot {i}\!}u_{n_{k}} \rightharpoonup u\) in D(Δ λ ). Using (21), we have

$$\|u_{n_{k}}-u\|^{2}_{{\overset{\circ}{W}}{^{1,2}_{\lambda}}{({\Omega})}}\leq \varepsilon \|{\Delta}_{\lambda} u_{n_{k}}-{\Delta}_{\lambda} u\|^{2}_{L^{2}({\Omega})}+C(\varepsilon)\|u_{n_{k}}-u\|^{2}_{L^{1}({\Omega})}. $$

Since \(\phantom {\dot {i}\!}D({\Delta }_{\lambda })\subset {\overset {\circ }{W}}{^{1,2}_{\lambda }}{({\Omega })}\subset \subset L^{1}({\Omega })\) and the boundedness of the sequence \(\phantom {\dot {i}\!}\{u_{n_{k}}-u\}\) in D(Δ λ ), we conclude that \(\phantom {\dot {i}\!}u_{n_{k}}\to u\) in \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}{({\Omega })}\), up to a subsequence if necessary. This completes the proof. □

As a direct consequence of Lemma 3 and the compactness of the embedding \(D({\Delta }_{\lambda }) \hookrightarrow {\overset {\circ }{W}}{^{1,2}_{\lambda }}{({\Omega })}\), we get the main result of this section.

Theorem 2

Suppose (F)–(G) hold. Then the semigroup S(t) generated by problem (1) has a compact global attractor \(\phantom {\dot {i}\!}\mathcal {A}\) in the space \(\phantom {\dot {i}\!}{\overset {\circ }{W}}{^{1,2}_{\lambda }}{({\Omega })}\).