Attractors and asymptotic regularity for nonclassical diffusion equations in locally uniform spaces with critical exponent

Abstract

In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations in locally uniform spaces $\begin{matrix} u_{t} - Δ u_{t} - Δ u + f (u) = g (x), x \in R^{N} . \end{matrix}$ First, we prove the well-posedness of solution for the nonclassical diffusion equations with critical nonlinearity in locally uniform spaces, and then the existence of $(H_{lu}^{1} (R^{N}), H_{ρ}^{1} (R^{N}))$ -global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appears to be optimal and the existence of a bounded (in $(H_{lu}^{2} (R^{N}))$ ) subset which attracts exponentially every initial $H_{lu}^{1} (R^{N})$ -bounded set with respect to the $H_{lu}^{1} (R^{N})$ -norm.

Keywords

nonclassical diffusion equations global attractor asymptotic regularity critical exponent locally uniform spaces

1. Introduction

The study of nonlinear dynamics is a fascinating question which is at the very heart of understanding of many important problems of the natural sciences. The long-time behavior of PDEs can be described in the terms of attractors of the corresponding semigroups, e.g. see Babin and Vishik [6,7], Robinson [22], Temam [24] and the references therein.

In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations in locally uniform spaces: $\begin{matrix} (1.1) & u_{t} - Δ u_{t} - Δ u + f (u) = g (x), x \in R^{N}, \end{matrix}$ with the initial data $\begin{matrix} (1.2) & u (x, 0) = u_{0}, x \in R^{N}, \end{matrix}$ where $g (x) \in L_{lu}^{2} (R^{N})$ with $N ⩾ 3$ , and the nonlinearity $f (u) \in C^{1} (R)$ satisfies the following conditions:

Dissipative condition $\begin{matrix} (1.3) & \underset{| s | \to \infty}{lim inf} \frac{F (s)}{s^{2}} ⩾ 0, where F (s) = \int_{0}^{s} f (r) d r, \end{matrix}$ $\begin{matrix} (1.4) & \underset{| s | \to \infty}{lim inf} \frac{s f (s) - α F (s)}{s^{2}} ⩾ 0, where α > 0 . \end{matrix}$

Growth condition $\begin{matrix} (1.5) & | f (s) - f (h) | ⩽ β | s - h | (1 + | s |^{q} + | h |^{q}), \forall s, h \in R, where β > 0, 0 ⩽ q ⩽ \frac{4}{N - 2} . \end{matrix}$

Nonclassical diffusion equations arise as models to describe physical phenomena such as non-Newtonian flow, soil mechanics, heat conduction, etc. (see Aifantis [1,2]; Kuttler and Aifantis [17,18] and references therein). In the work of Aifantis et al. [1,2], we can find a quite general approach to deduce these equations. In the these papers, they pointed out that the classical reaction–diffusion equation $\begin{matrix} u_{t} - Δ u = g (u), \end{matrix}$ does not contain each aspect of the reaction–diffusion problem, and it neglects viscidity, elasticity, and pressure of medium in the process of solid diffusion. The authors obtained a diffusion theory similar to Fick’s classical model for solute in an undisturbed solid matrix, obtaining a hyperbolic equation $\begin{matrix} u_{t} + D_{1} u_{t t} = D_{2} Δ u, \end{matrix}$ where $D_{1}$ and $D_{2}$ are positive constants. Assigning viscosity to the diffusing substance, they arrived at the following equation $\begin{matrix} u_{t} + D_{1} u_{t t} = D_{2} Δ u + D_{3} Δ u_{t} \end{matrix}$ and neglecting the inertia term, finally obtained the nonclassical parabolic equation $\begin{matrix} u_{t} = D_{2} Δ u + D_{3} Δ u_{t}, \end{matrix}$ where $D_{3}$ is also a positive constant.

The longtime behavior, especially the global attractor, exponential attractors of Eq. (1.1) acted on a bounded domain $Ω \subset R^{N}$ has been extensively studied by several authors in [16,19,23,27,28,31] and references therein. It is a nature extension is to consider problem (1.1) in unbounded domains $R^{N}$ . As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense. For bounded domains, the compactness is obtained by a priori estimates and compactness of Sobolev embeddings. This method does not apply to unbounded domains since the embeddings are no longer compact. To overcome the difficulty of the noncompact embedding, some authors in [3,20,26] use the “tail-estimate” method which was introduced by Wang [25] to obtain the existence of global (pullback) attractor for Eq. (1.1) with subcritical exponent (i.e., $q < \frac{4}{N - 2}$ ).

To our best knowledge, the long-time behavior for the Eqs (1.1)–(1.2) with critical exponent (i.e., $q ⩽ \frac{4}{N - 2}$ ) have not been considered by predecessors. Our main aim of this paper is to give a detailed analysis of the dynamics for Eqs (1.1)–(1.2) in locally uniform spaces $H_{lu}^{1} (R^{N})$ , including the global well-posedness, global attractor, asymptotic regularity of the solutions, and some exponential attraction with critical exponent. There are some barriers encountered. On the one hand, Eq. (1.1) contains the term $- Δ u_{t}$ , it is different from the usual reaction diffusion equation. For example, the reaction diffusion equation has some smoothing effect, e.g., although the initial data only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for Eq. (1.1), if the initial data $u_{0}$ belongs to $H_{lu}^{1} (R^{N})$ , then the solution $u (t, x)$ is always in $H_{lu}^{1} (R^{N})$ and has no higher regularity because of $- Δ u_{t}$ , it will cause some difficulties. On the other hand, the unbounded domain also brings some difficulties since the embeddings are no longer compact, so the asymptotic compactness of solutions can not be obtained by the standard method. Thirdly, the number $q = \frac{4}{N - 2}$ in (1.5) is called a critical exponent, since the nonlinearity f is not compact in this case (i.e., for a bounded subset $B \subset H_{lu}^{1} (R^{N})$ , in general, the set $f (B)$ is not precompact in $L_{ρ}^{\frac{2 N}{N + 2}} (R^{N})$ ).

This paper is organized as following: In Section 2, we recall some basic definitions about the locally uniform spaces that will be used later. In Section 3, we prove the existence of weak solution for a nonclassical diffusion equations. In Section 4, we prove the existence of global attractors. The asymptotic regularity result and exponential attraction are stated and proved in Section 5.

2. Preliminaries

In this section, we recall some basic definitions about the locally uniform spaces.

Following [4,5,9,10], we consider a strictly positive integrable weighted function $ρ : R^{N} \to (0, \infty)$ : for $1 ⩽ p < \infty$ , setting $\begin{matrix} L_{ρ}^{p} (R^{N}) = {φ \in L_{loc}^{p} (R^{N}) : ‖ φ ‖_{L_{ρ}^{p} (R^{N})} = {(\int_{R^{N}} ρ (x) {| φ (x) |}^{p} d x)}^{\frac{1}{p}} < \infty}, \end{matrix}$ let $τ_{y} ρ (x) = ρ_{y} (x) = ρ (x - y)$ , $y \in R^{N}$ , and consider the locally uniform spaces $\begin{array}{l} L_{lu}^{p} (R^{N}) = {φ \in L_{loc}^{p} (R^{N}) : ‖ φ ‖_{L_{lu}^{p} (R^{N})} = sup_{y \in R^{N}} ‖ φ ‖_{L_{ρ_{y}}^{p} (R^{N})} < \infty}, \\ {\dot{L}}_{lu}^{p} (R^{N}) = {φ \in L_{lu}^{p} (R^{N}) : ‖ τ_{y} φ - φ ‖_{L_{lu}^{p} (R^{N})} \to 0 as | y | \to 0}, \end{array}$ where ${\dot{L}}_{lu}^{p} (R^{N})$ is the closed subspace of $L_{lu}^{p} (R^{N})$ consisting of all its elements that are translation continuous. The locally uniform Sobolev spaces $W_{lu}^{m, p} (R^{N})$ and ${\dot{W}}_{lu}^{m, p} (R^{N})$ are defined, respectively, by $L_{lu}^{p} (R^{N})$ and ${\dot{L}}_{lu}^{p} (R^{N})$ in a way similar to the standard $W_{lu}^{m, p} (R^{N})$ .

We consider strictly positive integrable weighted functions $ρ \in C^{2} (R^{N})$ satisfying $\begin{matrix} (2.1) & | \frac{\partial ρ}{\partial x_{j}} (x) | ⩽ ρ_{0} ρ (x), | \frac{\partial^{2} ρ}{\partial x_{j} \partial x_{k}} (x) | ⩽ ρ_{1} ρ (x), \forall x \in R^{N}, j, k = 1, 2, \dots, N, \end{matrix}$ with certain positive constants $ρ_{0}$ , $ρ_{1}$ . In this paper, we consider the exemplary weighted functions $\begin{matrix} (2.2) & ρ (x) = {(1 + ϵ | x |^{2})}^{- s}, with s > \frac{N}{2}, ϵ > 0 . \end{matrix}$ Obviously $ρ \in C^{2} (R^{N})$ , one can obtain the estimates that $| \nabla ρ | ⩽ C \sqrt{ϵ} ρ$ and $| Δ ρ | ⩽ ϵ C ρ$ .

Now, we recall the uniform space $W_{U}^{s, p} (R^{N})$ , $s \in R^{+} \cup {0}$ , the Banach space consisting of all $ϕ \in W_{loc}^{s, p} (R^{N})$ such that $\begin{matrix} (2.3) & ‖ ϕ ‖_{W_{U}^{s, p} (R^{N})} = sup_{y \in R^{N}} ‖ ϕ ‖_{W_{U}^{s, p} (B (y, 1))} < \infty, \end{matrix}$ where $B (y, 1) = {x \in R^{N} : | x - y | ⩽ 1}$ . From [5,8], we know that for $k \in N \cup {0}$ , uniform space $W_{U}^{k, p} (R^{N})$ and locally uniform space $W_{lu}^{k, p} (R^{N})$ coincide algebraically and topologically when the weighted function ρ satisfies (2.1). Furthermore, by intermediate spaces we know that the same holds for $W_{U}^{s, p} (R^{N})$ and $W_{lu}^{s, p} (R^{N})$ with $s \in R^{+} \cup {0}$ , and we will use this equivalence frequently in this paper.

In addition, we need the following embedding lemma, interpolation inequalities in the weighted spaces and locally uniform space.

Lemma 2.1 ([5]).

(i) If $s_{1} ⩾ s_{2} ⩾ 0$ , $1 < p_{1} ⩽ p_{2} < \infty$ and $s_{1} - \frac{N}{p_{1}} ⩾ s_{2} - \frac{N}{p_{2}}$ , then $\begin{matrix} W_{U}^{s_{1}, p_{1}} (R^{N}) ↪ W_{U}^{s_{2}, p_{2}} (R^{N}) \end{matrix}$ is continuous.

(ii) If ρ satisfies ( 2.1 ), then the inclusion $\begin{matrix} W_{U}^{s_{1}, p_{1}} (R^{N}) ↪ W_{ρ}^{s_{2}, p_{2}} (R^{N}) \end{matrix}$ provided that $s_{2} \in N$ , $s_{1} > s_{2}$ , $1 < p_{1} ⩽ p_{2} < \infty$ and $s_{1} - \frac{N}{p_{1}} > s_{2} - \frac{N}{p_{2}}$ .

Lemma 2.2 ([5]).

For any $p \in [2, \frac{2 N}{N - 2}]$ and $θ \in [0, 1]$ , we have $\begin{matrix} ‖ φ ‖_{L_{ρ}^{p}} ⩽ C ‖ φ ‖_{H_{lu}^{1}}^{θ} ‖ φ ‖_{L_{ρ}^{r}}^{1 - θ} \end{matrix}$ and $\begin{matrix} ‖ φ ‖_{L_{ρ}^{p}} ⩽ C ‖ φ ‖_{H_{ρ}^{1}}^{θ} ‖ φ ‖_{L_{lu}^{r}}^{1 - θ} \end{matrix}$ where $\frac{1}{p} ⩽ \frac{θ}{2} + \frac{1 - θ}{r}$ and $- \frac{N}{p} ⩽ θ (1 - \frac{N}{2}) - (1 - θ) \frac{N}{r}$ .

We also need the following attraction transitivity lemma.

Lemma 2.3 ([15]).

Let $K_{1}$ , $K_{2}$ , $K_{3}$ be subsets of H such that $\begin{matrix} {dist}_{H} (S (t) K_{1}, K_{2}) ⩽ L_{1} e^{- ν_{1} t}, {dist}_{H} (S (t) K_{2}, K_{3}) ⩽ L_{2} e^{- ν_{2} t}, \end{matrix}$ for some $ν_{1}, ν_{2} > 0$ and $L_{1}, L_{2} > 0$ . Assume that for all $z_{1}, z_{2} \in ⋃_{t ⩾ 0} S (t) K_{j}$ ( $j = 1, 2, 3$ ) there holds $\begin{matrix} ‖ S (t) z_{1} - S (t) z_{2} ‖ ⩽ L_{0} e^{ν_{0} t} ‖ z_{1} - z_{2} ‖ \end{matrix}$ for some $ν_{0} > 0$ and some $L_{0} > 0$ . Then it follows that $\begin{matrix} {dist}_{H} (S (t) K_{1}, K_{3}) ⩽ L e^{- ν t}, \end{matrix}$ where $ν = \frac{ν_{1} ν_{2}}{ν_{0} + ν_{1} + ν_{2}}$ and $L = L_{0} L_{1} + L_{2}$ .

3. Well-posedness

In this section, we will consider the well-posedness for the problem (1.1)–(1.2).

Theorem 3.1.
Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ . Then for any $T > 0$ and $u_{0} \in H_{lu}^{1} (R^{N})$ , there is a unique solution u of Eqs ( 1.1 )–( 1.2 ) such that $\begin{matrix} u \in C^{1} ([0, T]; H_{lu}^{1} (R^{N})) . \end{matrix}$ Moreover, the solution continuously depends on the initial data.
Proof.
We divide into two steps: Step 1.
For any $n \in N$ , we consider the existence of the weak solution for the following problem in $B (0, n) ≜ B_{n} \subset R^{N}$ , $\begin{array}{l} (3.1) & u_{t} - Δ u_{t} - Δ u + f (u) = g (x), x \in B_{n}, \\ (3.2) & u (x, 0) = u_{0} \in H_{lu}^{1} (B_{n}) . \\ (3.3) & u |_{\partial Ω} = 0 . \end{array}$ Choose a smooth function $χ_{n} (x)$ with $\begin{matrix} (3.4) & χ_{n} (x) = \{\begin{matrix} 1, & x \in B_{n - 1}, \\ 0, & x \notin B_{n} . \end{matrix} \end{matrix}$ Since $B_{n}$ is a bounded domain, so the existence and uniqueness of solutions can be obtained by the standard Faedo–Galerkin methods, see [16,22–24,27], we have the unique weak solution $\begin{matrix} u_{n} \in C^{1} ([0, T]; H_{lu}^{1} (B_{n})) and u_{n} (x, 0) = χ_{n} (x) u_{0} (x) . \end{matrix}$
Step 2.
According to Step 1, and we denote $\frac{d}{d t} u_{n} = u_{n t}$ , then $u_{n}$ satisfy $\begin{array}{l} (3.5) & u_{n t} - Δ u_{n t} - Δ u_{n} + f (u_{n}) = g (x), x \in B_{n}, \\ (3.6) & u_{n} (x, 0) = χ_{n} (x) u_{0} (x), \\ (3.7) & u_{n} |_{\partial B_{n}} = 0 . \end{array}$ Multiplying (3.5) by $ρ_{y} (u_{n t} + \frac{1}{2} u_{n})$ in $B_{n}$ , we infer $\begin{array}{rcl} ⟨ u_{n t}, ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ - ⟨ Δ u_{n t}, ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ - ⟨ Δ u_{n}, ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ \\ (3.8) & + ⟨ f (u_{n}), ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ = ⟨ g (x), ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ . \end{array}$ Next, we deal with each term of (3.8) one by one as follows: $\begin{array}{l} (3.9) & ⟨ u_{n t}, ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ = \int_{B_{n}} ρ_{y} | u_{n t} |^{2} + \frac{1}{4} \frac{d}{d t} \int_{B_{n}} ρ_{y} | u_{n} |^{2}, \\ (3.10) & \begin{matrix} ⟨ - Δ u_{n t}, ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ = & \int_{B_{n}} ρ_{y} | \nabla u_{n t} |^{2} + \frac{1}{4} \frac{d}{d t} \int_{B_{n}} ρ_{y} | \nabla u_{n} |^{2} \\ + \int_{B_{n}} \nabla u_{n t} \nabla ρ u_{n t} + \frac{1}{2} \int_{B_{n}} \nabla u_{n t} \nabla ρ u_{n}, \end{matrix} \\ (3.11) & \begin{matrix} ⟨ - Δ u_{n}, ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ = & \frac{1}{2} \int_{B_{n}} \frac{d}{d t} ρ_{y} | \nabla u_{n} |^{2} + \frac{1}{2} \int_{B_{n}} ρ_{y} | \nabla u_{n} |^{2} \\ + \int_{B_{n}} \nabla u_{n} \nabla ρ u_{n t} + \frac{1}{2} \int_{B_{n}} \nabla u_{n} \nabla ρ u_{n}, \end{matrix} \\ (3.12) & ⟨ f (u_{n}), ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ = \frac{d}{d t} \int_{B_{n}} ρ_{y} F (u_{n}) + \frac{1}{2} \int_{B_{n}} ρ_{y} f (u_{n}) u_{n}, \\ (3.13) & ⟨ g (x), ρ_{y} (u_{n t} + \frac{1}{2} u_{n}) ⟩ ⩽ \frac{ε}{4} \int_{B_{n}} ρ_{y} (| u_{n t} |^{2} + | u_{n} |^{2}) + C_{ε} \int_{B_{n}} ρ_{y} | g |^{2} . \end{array}$ From (3.8)–(3.13), we get $\begin{array}{rcl} \frac{1}{4} \frac{d}{d t} \int_{B_{n}} ρ_{y} (| u_{n} |^{2} + 3 | \nabla u_{n} |^{2} + 4 F (u_{n})) + (1 - \frac{ε}{4}) \int_{B_{n}} ρ_{y} (| u_{n t} |^{2} + | \nabla u_{n t} |^{2}) \\ + \frac{1}{2} \int_{B_{n}} ρ_{y} | \nabla u_{n} |^{2} + \frac{1}{2} \int_{B_{n}} ρ_{y} f (u_{n}) u_{n} \\ ⩽ C_{ε} \int_{B_{n}} ρ_{y} | g |^{2} + \int_{B_{n}} \nabla u_{n t} \nabla ρ u_{n t} + \frac{1}{2} \int_{B_{n}} \nabla u_{n t} \nabla ρ u_{n} \\ (3.14) & + \int_{B_{n}} \nabla u_{n} \nabla ρ u_{n t} + \frac{1}{2} \int_{B_{n}} \nabla u_{n} \nabla ρ u_{n} . \end{array}$ By (1.3)–(1.4) (e.g. see [22,24]), we have $\begin{array}{rcl} (3.15) & \int_{B_{n}} ρ_{y} f (u_{n}) u_{n} ⩾ c_{1} \int_{B_{n}} ρ_{y} F (u_{n}) + μ \int_{B_{n}} ρ_{y} | u_{n} |^{2} - C_{μ} \int_{B_{n}} ρ_{y}, \end{array}$ and $\begin{array}{rcl} (3.16) & \int_{B_{n}} ρ_{y} F (u_{n}) ⩾ - c_{2} \int_{B_{n}} ρ_{y} . \end{array}$ Noting that $\begin{array}{rcl} (3.17) & \int_{B_{n}} \nabla u_{n t} \nabla ρ u_{n t} + \frac{1}{2} \int_{B_{n}} \nabla u_{n t} \nabla ρ u_{n} ⩽ C \sqrt{ϵ} (| \nabla u_{n t} |^{2} + | u_{n t} |^{2} + | u_{n} |^{2}), \\ (3.18) & \int_{B_{n}} \nabla u_{n} \nabla ρ u_{n t} + \frac{1}{2} \int_{B_{n}} \nabla u_{n} \nabla ρ u_{n} ⩽ C ϵ (| \nabla u_{n} |^{2} + | u_{n t} |^{2} + | u_{n} |^{2}) . \end{array}$ Substituting the estimates (3.15)–(3.18) into (3.14), choosing ϵ and ε small enough, and denoting $\begin{array}{rcl} (3.19) & E_{n} (t) = \int_{B_{n}} ρ_{y} (| u_{n} |^{2} + 3 | \nabla u_{n} |^{2} + 4 F (u_{n})), \end{array}$ we can obtain that $\begin{array}{rcl} (3.20) & \frac{d}{d t} E_{n} (t) + C_{ϵ, ε} E_{n} (t) ⩽ C \int_{B_{n}} ρ_{y} (| g |^{2} + 1) . \end{array}$ Using the Gronwall lemma, we infer $\begin{array}{rcl} (3.21) & E_{n} (t) ⩽ e^{- C_{ϵ, ε}} E_{n} (0) + C \int_{B_{n}} ρ_{y} (| g |^{2} + 1) . \end{array}$ Combing (3.14) with (3.21), we have $\begin{array}{rcl} (3.22) & \int_{0}^{T} \int_{B_{n}} ρ_{y} (| u_{n t} |^{2} + | \nabla u_{n t} |^{2}) ⩽ C . \end{array}$

Note that $u_{n}$ is the solution of Eq. (1.1) with the initial data $u_{n} (x, 0) (= χ_{n} (x) u_{0} (x))$ , if we define $\begin{matrix} u_{n} = 0, \forall x \in R^{N} ∖ B_{n}, \end{matrix}$ then we can extend the solution $u_{n}$ to $R^{N}$ .

Hence, for any $T > 0$ and $n \in N$ , Eq. (3.21) holds true, and there exists a subsequence ${u_{n_{j}}}$ satisfies $\begin{matrix} n_{j} \to + \infty, as j \to + \infty, \end{matrix}$ and $\begin{array}{l} u_{n_{j}} \to u_{\infty} weakly in L^{2} ([0, T]; H_{lu}^{1} (R^{N})), \\ u_{n_{j}} \to u_{\infty} star-weakly in L^{\infty} ([0, T]; H_{lu}^{1} (R^{N})), \\ \partial_{t} u_{n_{j}} \to \partial_{t} u_{\infty} weakly in L^{2} ([0, T]; H_{lu}^{1} (R^{N})), \end{array}$ where $u_{\infty} = {lim}_{j \to + \infty} u_{n_{j}}$ . Now, similar to the analysis in [6,7,19,22,24,25], we know that $u = u_{\infty}$ is a weak solution and satisfies $u \in C^{1} ([0, T]; H_{lu}^{1} (R^{N}))$ .

The uniqueness and the continuous dependence on initial conditions can be proof in a similar way, we omit the proof here. □
Remark 3.2.
In fact, we can also prove the existence of solutions in the framework of the global solvability. Step 1 (Local well-posedness).

Note that Eq. (1.1) can be written as $\begin{array}{rcl} u_{t} & = & {(I - Δ)}^{- 1} [Δ u - u + u - f (u) + g (x)] \\ (3.23) & = & - u + {(I - Δ)}^{- 1} [u - f (u) + g (x)] . \end{array}$ Moreover, the inverse operator ${(I - Δ)}^{- 1}$ is a sectorial positive operator, and it has nice regularizing properties (e.g. [11,12,14,27]). By Eq. (1.5), we know that $H (u) = u - f (u) + g (x)$ is a local Lipschitz function, then similar to the arguments in [11,12,14], the solution of Eq. (1.1) satisfying the following Cauchy’s integral formula: $\begin{matrix} u (t) = e^{- t} u_{0} + \int_{0}^{t} e^{- t - s} ({(I - Δ)}^{- 1} [u - f (u) + g (x)]) . \end{matrix}$

Step 2 (Global existence).

By the a priori estimate given in Theorem 4.1 (e.g. Eq. (4.15)), we know that for each local solution $u (t)$ corresponding to the initial data $u_{0} \in H_{lu}^{1} (R^{N})$ , its $H_{lu}^{1}$ -norm can not blow up in finite time, which implies the global existence of solutions.

Remark 3.3.
Theorem 3.1 implies that the solution of Eqs (1.1)–(1.2) generates a $C^{0}$ semigroup ${S (t)}_{t ⩾ 0}$ in the space $H_{lu}^{1} (R^{N})$ .

4. Global attractor

In the section, we will prove the existence of global attractor for a nonclassical diffusion equations.

4.1. Dissipation estimates

Theorem 4.1.
Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ . There is a positive constant $ϱ_{1}$ such that for any bounded subset $B \subset {\dot{W}}_{lu}^{1, 2} (R^{N})$ , there exists a positive constant $T_{1} = T_{1} (B)$ such that $\begin{array}{rcl} (4.1) & {‖ u (t) ‖}_{H_{lu}^{1}} ⩽ ϱ_{1} for all t ⩾ T_{1} and u_{0} \in B . \end{array}$
Proof.
Multiplying (1.1) by $ρ_{y} (u_{t} + \frac{1}{2} u)$ in $B_{n}$ , we infer $\begin{array}{rcl} ⟨ u_{t}, ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ - ⟨ Δ u_{t}, ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ - ⟨ Δ u, ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ \\ (4.2) & + ⟨ f (u), ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ = ⟨ g (x), ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ . \end{array}$ Next, we deal with each term of (4.2) one by one as follows: $\begin{array}{l} (4.3) & ⟨ u_{t}, ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ = \int_{R^{N}} ρ_{y} | u_{t} |^{2} + \frac{1}{4} \frac{d}{d t} \int_{R^{N}} ρ_{y} | u |^{2}, \\ (4.4) & \begin{matrix} ⟨ - Δ u_{t}, ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ = & \int_{R^{N}} ρ_{y} | \nabla u_{t} |^{2} + \frac{1}{4} \frac{d}{d t} \int_{R^{N}} ρ_{y} | \nabla u |^{2} \\ + \int_{R^{N}} \nabla u_{t} \nabla ρ u_{t} + \frac{1}{2} \int_{R^{N}} \nabla u_{t} \nabla ρ u, \end{matrix} \\ (4.5) & \begin{matrix} ⟨ - Δ u, ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ = & \frac{1}{2} \int_{R^{N}} \frac{d}{d t} ρ_{y} | \nabla u |^{2} + \frac{1}{2} \int_{R^{N}} ρ_{y} | \nabla u |^{2} \\ + \int_{R^{N}} \nabla u \nabla ρ u_{t} + \frac{1}{2} \int_{R^{N}} \nabla u \nabla ρ u, \end{matrix} \\ (4.6) & ⟨ f (u), ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ = \frac{d}{d t} \int_{R^{N}} ρ_{y} F (u) + \frac{1}{2} \int_{R^{N}} ρ_{y} f (u) u, \\ (4.7) & ⟨ g (x), ρ_{y} (u_{t} + \frac{1}{2} u) ⟩ ⩽ \frac{ε}{4} \int_{R^{N}} ρ_{y} (| u_{t} |^{2} + | u |^{2}) + C_{ε} \int_{R^{N}} ρ_{y} | g |^{2} . \end{array}$ From (4.2)–(4.7), we get $\begin{array}{l} \frac{1}{4} \frac{d}{d t} \int_{R^{N}} ρ_{y} (| u |^{2} + 3 | \nabla u |^{2} + 4 F (u)) + (1 - \frac{ε}{4}) \int_{R^{N}} ρ_{y} (| u_{t} |^{2} + | \nabla u_{t} |^{2}) \\ + \frac{1}{2} \int_{R^{N}} ρ_{y} | \nabla u |^{2} + \frac{1}{2} \int_{R^{N}} ρ_{y} f (u) u \\ ⩽ C_{ε} \int_{R^{N}} ρ_{y} | g |^{2} + \int_{R^{N}} \nabla u_{t} \nabla ρ u_{t} + \frac{1}{2} \int_{R^{N}} \nabla u_{t} \nabla ρ u \\ (4.8) & + \int_{R^{N}} \nabla u \nabla ρ u_{t} + \frac{1}{2} \int_{R^{N}} \nabla u \nabla ρ u . \end{array}$ By (1.3)–(1.4), we infer $\begin{matrix} (4.9) & \int_{R^{N}} ρ_{y} f (u) u ⩾ c_{1} \int_{R^{N}} ρ_{y} F (u) + μ \int_{R^{N}} ρ_{y} | u |^{2} - C_{μ} \int_{R^{N}} ρ_{y}, \end{matrix}$ and $\begin{array}{rcl} (4.10) & \int_{R^{N}} ρ_{y} F (u) ⩾ - c_{2} \int_{R^{N}} ρ_{y} . \end{array}$ Noting that $\begin{array}{l} (4.11) & \int_{R^{N}} \nabla u_{t} \nabla ρ u_{t} + \frac{1}{2} \int_{R^{N}} \nabla u_{t} \nabla ρ u ⩽ C \sqrt{ϵ} (| \nabla u_{t} |^{2} + | u_{t} |^{2} + | u |^{2}), \\ (4.12) & \int_{R^{N}} \nabla u \nabla ρ u_{t} + \frac{1}{2} \int_{R^{N}} \nabla u \nabla ρ u ⩽ C ϵ (| \nabla u |^{2} + | u_{t} |^{2} + | u |^{2}) . \end{array}$ Substituting the estimates (4.9)–(4.12) into (4.8), choosing ϵ and ε small enough, and denoting $\begin{array}{rcl} (4.13) & E (t) = \int_{R^{N}} ρ_{y} (| u |^{2} + 3 | \nabla u |^{2} + 4 F (u)), \end{array}$ we can obtain that $\begin{array}{rcl} (4.14) & \frac{d}{d t} E (t) + C_{ϵ, ε} E (t) ⩽ C \int_{R^{N}} ρ_{y} (| g |^{2} + 1) . \end{array}$ Using the Gronwall lemma, we infer $\begin{array}{rcl} (4.15) & E (t) ⩽ e^{- C_{ϵ, ε}} E (0) + C \int_{R^{N}} ρ_{y} (| g |^{2} + 1) . \end{array}$

This completes the proof. □
Remark 4.2.
Theorem 4.1 implies that the $C^{0}$ semigroup ${S (t)}_{t ⩾ 0}$ has a bounded absorbing set in the locally uniform space $H_{lu}^{1} (R^{N})$ .
Lemma 4.1.
Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ . There is a positive constant $ϱ_{2}$ such that for any bounded subset $B \subset {\dot{W}}_{lu}^{1, 2} (R^{N})$ , there exists a positive constant $T_{2} = T_{2} (B)$ such that $\begin{matrix} (4.16) & {‖ u_{t} (t) ‖}_{H_{lu}^{1}} ⩽ ϱ_{2} for all t ⩾ T_{2} and u_{0} \in B . \end{matrix}$
Proof.
Multiplying (1.1) by $ρ_{y} u_{t}$ , we infer $\begin{array}{l} \int_{R^{N}} (| u_{t} |^{2} + | \nabla u_{t} |^{2}) = & - \int_{R^{N}} \nabla ρ_{y} \nabla u_{t} u_{t} - \int_{R^{N}} ρ_{y} \nabla u_{t} \nabla u \\ (4.17) & - \int_{R^{N}} \nabla ρ_{y} \nabla u u_{t} - \int_{R^{N}} ρ_{y} f (u) u_{t} + \int_{R^{N}} ρ_{y} g u_{t} . \end{array}$ Next, we deal with each term of (4.17) one by one as follows: $\begin{array}{l} (4.18) & | \int_{R^{N}} \nabla ρ_{y} \nabla u_{t} u_{t} | ⩽ C \sqrt{ϵ} \int_{R^{N}} ρ_{y} (| \nabla u_{t} |^{2} + | u_{t} |^{2}), \\ (4.19) & | \int_{R^{N}} ρ_{y} \nabla u_{t} \nabla u | ⩽ ϵ \int_{R^{N}} ρ_{y} | \nabla u_{t} |^{2} + C ‖ u ‖_{{\dot{H}}_{lu}^{1}}^{2}, \\ (4.20) & | \int_{R^{N}} \nabla ρ_{y} \nabla u u_{t} | ⩽ ϵ \int_{R^{N}} ρ_{y} | u_{t} |^{2} + C ‖ u ‖_{{\dot{H}}_{lu}^{1}}^{2}, \\ (4.21) & | \int_{R^{N}} ρ_{y} g u_{t} | ⩽ ϵ \int_{R^{N}} ρ_{y} | u_{t} |^{2} + C ‖ g ‖_{{\dot{L}}_{lu}^{2}}^{2}, \end{array}$ and $\begin{array}{rcl} | \int_{R^{N}} ρ_{y} f (u) u_{t} | & ⩽ & \int_{R^{N}} ρ_{y} (r) (\int_{B_{1} (r)} (1 + | u |^{\frac{N + 2}{N - 2}}) | u_{t} | d x) d r \\ ⩽ & \int_{R^{N}} ρ_{y} (r) {(\int_{B_{1} (r)} {(1 + | u |^{\frac{N + 2}{N - 2}})}^{\frac{2 N}{N + 2}})}^{\frac{N + 2}{2 N}} {(\int_{B_{1} (r)} | u_{t} |^{\frac{2 N}{N - 2}} d x)}^{\frac{N - 2}{2 N}} d r \\ ⩽ & \int_{R^{N}} ρ_{y} (r) {(\int_{B_{1} (r)} {(1 + | u |^{\frac{N + 2}{N - 2}})}^{\frac{2 N}{N + 2}})}^{\frac{N + 2}{2 N}} {(\int_{B_{1} (r)} | u_{t} |^{\frac{2 N}{N - 2}} d x)}^{\frac{N - 2}{2 N}} d r \\ ⩽ & \int_{R^{N}} ρ_{y} (r) ‖ u ‖_{H^{1} (B_{1} (r))} ‖ u_{t} ‖_{B_{1} (r)} d r \\ (4.22) & ⩽ & ϵ \int_{R^{N}} ρ_{y} | u_{t} |^{2} + C_{ϵ} ‖ u ‖_{{\dot{H}}_{lu}^{1}}^{2} . \end{array}$ Substituting the estimates (4.18)–(4.22) into (4.17), and choosing ϵ small enough, by (4.1), we complete the proof. □
4.2. Decomposition of the equations

For the nonlinear term f, following the idea in [13,21–24,29,30], for a $C^{1}$ -function satisfying (1.3)–(1.5), the following decomposing properties hold: there are constants $C > 0$ and γ satisfying $0 < γ < q + 1$ such that f can be decomposed as $\begin{matrix} f = f_{0} + f_{1} \end{matrix}$ with $f_{0}$ , $f_{1} \in C^{1} (R)$ satisfying $\begin{array}{rcl} (4.23) & | f_{0} (s) | ⩽ C (| s | + | s |^{q + 1}) \forall s \in R, \\ (4.24) & f_{0} (s) s ⩾ 0, \forall s \in R, \\ (4.25) & | f_{1} (s) | ⩽ C (1 + | s |^{γ}), \forall s \in R with some γ < q + 1, \\ (4.26) & \exists l \in R, F_{1} (s) ⩾ - l, \forall s \in R, \end{array}$ $\exists k_{i} ⩾ 1$ , ${\tilde{μ}}_{i} ⩾ 0$ such that $\forall μ_{i} \in (0, {\tilde{μ}}_{i}]$ , $\exists C_{μ_{i}} \in R$ , $\begin{matrix} (4.27) & k_{i} F_{i} (s) + μ_{i} s^{2} - C_{μ_{i}} ⩽ s f_{i} (s), for all s \in R, \end{matrix}$ where $F_{i} (s) = \int_{0}^{s} f_{i} (r) d r$ , $i = 1, 2$ .

Now we decompose the solution into the sum $\begin{matrix} S (t) u_{0} = D (t) u_{0} + K (t) u_{0} . \end{matrix}$ where $v = D (t) u_{0}$ and $w = K (t) u_{0}$ solves the following equations respectively: $\begin{matrix} (4.28) & \{\begin{matrix} v_{t} - Δ v_{t} - Δ v + f_{0} (v) = 0, \\ v (0) = u_{0}, \end{matrix} \end{matrix}$ and $\begin{matrix} (4.29) & \{\begin{matrix} w_{t} - Δ w_{t} - Δ w + f (u) - f_{0} (v) = g (x), \\ w (0) = 0 . \end{matrix} \end{matrix}$

Note that ${D (t)}_{t ⩾ 0}$ also forms a semigroup, but ${K (t)}_{t ⩾ 0}$ may not.

Lemma 4.2.
Assume that $f_{0}$ satisfies ( 4.23 )–( 4.24 ), ( 4.27 ). Then there exists a positive constant $k_{0}$ such that for every $t ⩾ T$ , $\begin{matrix} (4.30) & {‖ D (t) u_{0} ‖}_{H_{lu}^{1}}^{2} ⩽ Q_{1} (‖ u_{0} ‖_{H_{lu}^{1}}) e^{- k_{0} t}, \end{matrix}$ where $Q_{1} (\cdot)$ is an increasing function on $[0, \infty)$ .

Similar to the proof of Theorem 4.1, we can get Lemma 4.2 easily, so we omit the proof.
Lemma 4.3.
Assume that f satisfies ( 1.3 )–( 1.5 ), $f_{1}$ satisfies ( 4.25 )–( 4.27 ), $g (x) \in L_{lu}^{2} (R^{N})$ . Then there exists a positive constant $k_{1}$ such that for every $t ⩾ 0$ , $\begin{matrix} (4.31) & {‖ K (t) u_{0} ‖}_{H_{lu}^{1 + σ}}^{2} ⩽ Q_{2} (‖ u_{0} ‖_{H_{lu}^{1}}, ‖ g ‖_{L_{lu}^{2}}) e^{k_{1} t}, \end{matrix}$ where $Q_{2} (\cdot)$ is an increasing function on $[0, \infty)$ , $σ = min {\frac{1}{4}, \frac{N + 2 - (N - 2) γ}{2}}$ , where γ is given in ( 4.18 ).
Proof.
Let Θ be a smooth function satisfy $0 ⩽ θ (s) ⩽ 1$ for $s \in [0, \infty)$ , and $\begin{matrix} θ (s) = 1 for 0 ⩽ s ⩽ \frac{1}{2}; θ (s) = 0 for s ⩾ 1 . \end{matrix}$ Set $θ_{y} (x) = θ (| x - y |)$ and $A = - Δ$ . Multiplying (4.29) by $θ_{y} A^{σ} (θ_{y} w_{t})$ , we infer $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} + \int_{R^{N}} {| A^{\frac{σ}{2}} (θ_{y} w_{t}) |}^{2} + \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w_{t}) |}^{2} \\ + ⟨ f (u) - f (v) + f_{1} (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ \\ = ⟨ g, θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ - ⟨ Δ θ_{y} w_{t} + 2 \nabla θ_{y} \nabla w_{t}, A^{σ} (θ_{y} w_{t}) ⟩ \\ (4.32) & - ⟨ Δ θ_{y} w + 2 \nabla θ_{y} \nabla w, A^{σ} (θ_{y} w_{t}) ⟩ . \end{array}$ Note that $σ ⩽ \frac{N + 2 - (N - 2) γ}{2}$ , by (4.25), we get $\begin{array}{l} | ⟨ f_{1} (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ | \\ ⩽ C \int_{R^{N}} θ_{y} (1 + | v |^{γ}) | A^{σ} (θ_{y} w_{t}) | \\ ⩽ C {(\int_{R^{N}} {| A^{σ} (θ_{y} w_{t}) |}^{\frac{2 N}{N - 2 + 2 σ}})}^{\frac{N - 2 + 2 σ}{2 N}} {(\int_{B (y, 1)} {| 1 + | v |^{γ} |}^{\frac{2 N}{N + 2 - 2 σ}})}^{\frac{N + 2 - 2 σ}{2 N}} \\ (4.33) & ⩽ C (1 + ‖ v ‖_{H_{U}^{1}}^{γ}) (‖ θ_{y} w_{t} ‖_{L^{2}} + {‖ A^{\frac{σ + 1}{2}} (θ_{y} w_{t}) ‖}_{L^{2}}), \end{array}$ where $B (y, 1) = {x \in R^{N} : | x - y | ⩽ 1}$ .

By virtue of (1.5), we have $\begin{array}{l} | ⟨ f (u) - f (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ | \\ ⩽ C \int_{R^{N}} θ_{y} | w | (1 + | u |^{\frac{4}{N - 2}} + | v |^{\frac{4}{N - 2}}) | A^{σ} (θ_{y} w_{t}) | \\ ⩽ C {(\int_{R^{N}} {| A^{σ} (θ_{y} w_{t}) |}^{\frac{2 N}{N - 2 + 2 σ}})}^{\frac{N - 2 + 2 σ}{2 N}} {(\int_{R^{N}} | θ_{y} w |^{\frac{2 N}{N - 2 - 2 σ}})}^{\frac{N - 2 - 2 σ}{2 N}} \\ \times {(\int_{B (y, 1)} {| 1 + | u |^{\frac{4}{N - 2}} + | v |^{\frac{4}{N - 2}} |}^{\frac{N}{2}})}^{\frac{2}{N}} \\ (4.34) & ⩽ C (‖ θ_{y} w_{t} ‖_{L^{2}} + {‖ A^{\frac{σ + 1}{2}} (θ_{y} w_{t}) ‖}_{L^{2}}) (‖ θ_{y} w ‖_{L^{2}} + {‖ A^{\frac{σ + 1}{2}} (θ_{y} w) ‖}_{L^{2}}), \end{array}$ where $σ ⩽ \frac{1}{4} ⩽ \frac{N - 2}{2}$ and $B (y, 1) = {x \in R^{N} : | x - y | ⩽ 1}$ .

Since $σ < \frac{1}{2}$ , by the continuous embedding, we infer $\begin{array}{l} (4.35) & \begin{matrix} | ⟨ Δ θ_{y} w_{t} + 2 \nabla θ_{y} \nabla w_{t}, A^{σ} (θ_{y} w_{t}) ⟩ | \\ ⩽ C ‖ w_{t} ‖_{H_{U}^{1}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w_{t}) |}^{2})}^{\frac{1}{2}} \\ ⩽ ‖ w_{t} ‖_{H_{U}^{1}}^{2}, \end{matrix} \\ (4.36) & \begin{matrix} | ⟨ Δ θ_{y} w + 2 \nabla θ_{y} \nabla w, A^{σ} (θ_{y} w_{t}) ⟩ | \\ ⩽ C ‖ w ‖_{H_{U}^{1}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w_{t}) |}^{2})}^{\frac{1}{2}} \\ ⩽ ‖ w ‖_{H_{U}^{1}} ‖ w_{t} ‖_{H_{U}^{1}}, \end{matrix} \end{array}$ and $\begin{matrix} (4.37) & ⟨ g, θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ ⩽ ‖ g ‖_{L_{U}^{2}} ‖ w_{t} ‖_{H_{U}^{1}} . \end{matrix}$ Therefore, we have $\begin{matrix} (4.38) & \frac{d}{d t} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} ⩽ C \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} + C_{‖ g ‖_{L_{U}^{2}}, ‖ w_{t} ‖_{H_{U}^{1}}} . \end{matrix}$ Hence, applying the Gronwall lemma, we infer $\begin{array}{rcl} (4.39) & \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} ⩽ Q_{2} (‖ u ‖_{H_{lu}^{1}, ‖ g ‖_{L_{lu}^{2}}}) e^{k_{1} t} . \end{array}$ This complete the proof. □

Now, we state our main results: Theorem 4.3.
Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ . Then the semigroup ${S (t)}_{t ⩾ 0}$ generated by the weak solutions of ( 1.1 )–( 1.2 ) with the initial data $u_{0} \in H_{lu}^{1} (R^{N})$ has a unique $(H_{lu}^{1} (R^{N}), H_{ρ}^{1} (R^{N}))$ global attractor.

5. Regularity

In this section, we will prove the regularity of the $(H_{lu}^{1} (R^{N}), H_{ρ}^{1} (R^{N}))$ -global attractor by some bootstrap arguments. Similar to that Zelik [15,31], based on Lemmas 4.2 and 4.3, we can decompose $u (t)$ as follows.

Lemma 5.1.

Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ , and $u (t)$ be the solution of Eqs ( 1.1 )–( 1.2 ) with the initial data $u_{0} \in B$ . Then for any $ϵ > 0$ , there are positive constants $C_{ϵ}$ and $K_{ϵ}$ , such that $\begin{matrix} (5.1) & u (t) = v_{1} (t) + w_{1} (t), for all t ⩾ 0, \end{matrix}$ where $v_{1} (t)$ , $w_{1} (t)$ satisfy estimates as follows: $\begin{matrix} (5.2) & {‖ w_{1} (t) ‖}_{H_{U}^{1 + σ}}^{2} ⩽ K_{ϵ}, t ⩾ 0 \end{matrix}$ and for every $t ⩾ s ⩾ 0$ , $\begin{matrix} (5.3) & \int_{s}^{t} {‖ v_{1} (r) ‖}_{H_{U}^{1}}^{2} d r ⩽ ϵ (t - s) + C_{ϵ}, \end{matrix}$ where the constants $C_{ϵ}$ and $K_{ϵ}$ depending on ϵ, σ.

Proof.

Note that $\begin{matrix} B = ⋃_{t ⩾ T_{B_{0}}} S (t) B_{0}, \end{matrix}$ by Theorem 4.1, we infer $\begin{matrix} sup_{t ⩾ 0} {‖ S (t) u_{0} ‖}_{H_{lu}^{1}}^{2} ⩽ ϱ_{1}, for all u_{0} \in B . \end{matrix}$

Now, taking $T ⩾ \frac{1}{k_{0}} ln \frac{Q_{1} (ϱ_{1})}{ε}$ (where $Q_{1} (\cdot)$ the function in Lemma 4.2), and in every interval $[m T, (m + 1) T)$ , $m = 1, 2, \dots$ , we set $\begin{matrix} v_{1} = v (t) and w_{1} (t) = w (t), \end{matrix}$ where $v (t)$ is the solutions of (4.28) in the interval $[(m - 1) T, (m + 1) T)$ with the initial data $v ((m - 1) T) = u ((m - 1) T)$ , and $w (t)$ is the solutions of (4.29) in the interval $[(m - 1) T, (m + 1) T)$ with the initial data $w ((m - 1) T) = 0$ .

And in the interval $[0, T)$ we set $\begin{matrix} v_{1} = v (t) and w_{1} (t) = w (t), \end{matrix}$ where $v (t)$ is the solutions of (4.28) with the initial data $v (0) = u_{0}$ , and $w (t)$ is the solutions of (4.29) with the initial data $w (0) = 0$ .

Then from Lemma 4.2, we infer $\begin{matrix} \int_{s}^{t} {‖ v_{1} (r) ‖}_{H_{u}^{1}}^{2} d r ⩽ ε (t - s) + χ_{[0, T)} (s) Q_{1} (ϱ_{1}), for all t ⩾ s ⩾ 0, \end{matrix}$ where $χ_{[0, T)} (s)$ is the characteristic function of set $[0, T)$ . According to Lemma 4.3, we infer $\begin{matrix} {‖ w_{1} (t) ‖}_{H_{u}^{1 + σ}}^{2} ⩽ Q_{2} (‖ u_{0} ‖_{H_{lu}^{1}}, ‖ g ‖_{L_{lu}^{2}}) e^{2 k_{1} T}, for all t ⩾ 0 . \end{matrix}$

This complete the proof. □

Remark 5.1.

According to the proof of Lemma 5.1, we observe that the decomposition $v_{1}$ can also further satisfy that $\begin{matrix} {‖ v_{1} (t) ‖}_{H_{U}^{1}}^{2} ⩽ Q_{1} (ϱ), for all t ⩾ 0 . \end{matrix}$

Lemma 5.2.

Assume that f satisfies ( 1.3 )–( 1.5 ), $f_{1}$ satisfies ( 4.25 )–( 4.27 ), $g (x) \in L_{lu}^{2} (R^{N})$ . For any bounded set $B \subset H_{lu}^{1} (R^{N})$ , there exists a positive constant $J_{‖ B ‖_{H_{lu}^{1}}}$ which depends only on the $H_{lu}^{1}$ -bounds of B, such that $\begin{matrix} (5.4) & {‖ K_{g} (t) u_{0} ‖}_{H_{lu}^{1 + σ}}^{2} ⩽ J_{‖ B ‖_{H_{lu}^{1}}}, for all t ⩾ 0 and u_{0} \in B, \end{matrix}$ where σ is given in Lemma 4.3 .

Proof.

Multiplying (4.29) by $θ_{y} A^{σ} (θ_{y} w)$ , we infer $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} + \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} + ⟨ Δ θ_{y} w + 2 \nabla θ_{y} \nabla w, A^{σ} (θ_{y} w) ⟩ \\ + ⟨ Δ θ_{y} w_{t} + 2 \nabla θ_{y} \nabla w_{t}, A^{σ} (θ_{y} w) ⟩ + ⟨ f (u) - f (v) + f_{1} (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ \\ (5.5) & ⩽ ⟨ g, θ_{y} A^{σ} (θ_{y} w) ⟩ + ‖ w_{t} ‖_{L_{U}^{2}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w) |}^{2})}^{\frac{1}{2}}, \end{array}$ where $\begin{array}{l} (5.6) & ⟨ Δ θ_{y} w + 2 \nabla θ_{y} \nabla w, A^{σ} (θ_{y} w) ⟩ ⩽ C ‖ w ‖_{H_{U}^{1}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w) |}^{2})}^{\frac{1}{2}}, \\ (5.7) & ⟨ Δ θ_{y} w_{t} + 2 \nabla θ_{y} \nabla w_{t}, A^{σ} (θ_{y} w) ⟩ ⩽ C ‖ w_{t} ‖_{H_{U}^{1}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w) |}^{2})}^{\frac{1}{2}} . \end{array}$ By (1.5), we have $\begin{array}{l} ⟨ f (u) - f (v) + f_{1} (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ \\ = ⟨ f (u) - f (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ + ⟨ f_{1} (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ \\ ⩽ C \int_{R^{N}} (1 + | u |^{\frac{4}{N - 2}} + | v |^{\frac{4}{N - 2}}) θ_{y} | w | | A^{σ} (θ_{y} w) | + ⟨ f_{1} (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ \\ (5.8) & ≜ \sum_{i = 1}^{4} I_{i} . \end{array}$ For $I_{1}$ , we infer $\begin{array}{rcl} I_{1} & = & C \int_{R^{N}} θ_{y} | w | | A^{σ} (θ_{y} w) | \\ (5.9) & ⩽ & ‖ w ‖_{L_{U}^{2}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w) |}^{2})}^{\frac{1}{2}} . \end{array}$ For $I_{2}$ , using Lemma 5.1, we get $\begin{array}{rcl} I_{2} & = & C \int_{R^{N}} | u |^{\frac{4}{N - 2}} θ_{y} | w | | A^{σ} (θ_{y} w) | \\ (5.10) & ⩽ & C \int_{R^{N}} (| v_{1} |^{\frac{4}{N - 2}} + | w_{1} |^{\frac{4}{N - 2}}) θ_{y} | w | | A^{σ} (θ_{y} w) |, \end{array}$ by Remark 5.1 and the interpolation inequality, we have $\begin{array}{l} \int_{R^{N}} | v_{1} |^{\frac{4}{N - 2}} θ_{y} | w | | A^{σ} (θ_{y} w) | \\ = \int_{B (y, 1)} | v_{1} |^{\frac{4}{N - 2}} | θ_{y} w | | A^{σ} (θ_{y} w) | \\ ⩽ C {(\int_{B (y, 1)} | v_{1} |^{\frac{2 N}{N - 2}})}^{\frac{2}{N}} {(\int_{R^{N}} | θ_{y} w |^{\frac{2 N}{N - 2 - 2 σ}})}^{\frac{N - 2 - 2 σ}{2 N}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w) |}^{\frac{2 N}{N - 2 + 2 σ}})}^{\frac{N - 2 + 2 σ}{2 N}} \\ ⩽ C ‖ v_{1} ‖_{H_{U}^{1}}^{\frac{4}{N - 2}} ({‖ A^{\frac{1 + σ}{2}} (θ_{y} w) ‖}_{L^{2}} + ‖ θ_{y} w ‖_{L^{2}}) ({‖ A^{\frac{1 + σ}{2}} (θ_{y} w) ‖}_{L^{2}} + {‖ A^{σ} (θ_{y} w) ‖}_{L^{2}}) \\ (5.11) & ⩽ C_{ϱ_{1}} ‖ v_{1} ‖_{H_{U}^{1}}^{2} \int_{R^{N}} {| A^{\frac{1 + σ}{2}} (θ_{y} w) |}^{2} + \frac{1}{8} \int_{R^{N}} {| A^{\frac{1 + σ}{2}} (θ_{y} w) |}^{2} + C_{ϱ_{1}} (1 + ‖ θ_{y} w ‖_{L^{2}}^{2}), \end{array}$ by Lemma 5.1 and the interpolation inequality, we infer $\begin{array}{l} \int_{R^{N}} | w_{1} |^{\frac{4}{N - 2}} θ_{y} | w | | A^{σ} (θ_{y} w) | \\ ⩽ C {(\int_{B (y, 1)} | w_{1} |^{\frac{2 N}{N - 2 - 2 σ}})}^{\frac{2 (N - 2 - 2 σ)}{N (N - 2)}} {(\int_{R^{N}} | θ_{y} w |^{\frac{2 N (N - 2)}{{(N - 2)}^{2} - 2 (N - 6) σ}})}^{\frac{{(N - 2)}^{2} - 2 (N - 6) σ}{2 N (N - 2)}} \\ \times {(\int_{R^{N}} {| A^{σ} (θ_{y} w) |}^{\frac{2 N}{N - 2 + 2 σ}})}^{\frac{N - 2 + 2 σ}{2 N}} \\ ⩽ C ‖ w_{1} ‖_{H_{U}^{1 + σ}}^{\frac{4}{N - 2}} ‖ θ_{y} w ‖_{L^{\frac{2 N (N - 2)}{{(N - 2)}^{2} - 2 (N - 6) σ}}} ({‖ A^{\frac{1 + σ}{2}} (θ_{y} w) ‖}_{L^{2}} + {‖ A^{σ} (θ_{y} w) ‖}_{L^{2}}) \\ (5.12) & ⩽ \frac{1}{8} \int_{R^{N}} {| A^{\frac{1 + σ}{2}} (θ_{y} w) |}^{2} + C K_{ϵ} (1 + ‖ θ_{y} w ‖_{L^{2}}^{2}) . \end{array}$ Hence, $\begin{array}{rcl} I_{2} & ⩽ & \frac{1}{4} \int_{R^{N}} {| A^{\frac{1 + σ}{2}} (θ_{y} w) |}^{2} + C_{K_{ϵ}, ϱ_{1}} (1 + ‖ θ_{y} w ‖_{L^{2}}^{2}) \\ (5.13) & + C_{ϱ_{1}} ‖ v_{1} ‖_{H_{U}^{1}}^{2} \int_{R^{N}} {| A^{\frac{1 + σ}{2}} (θ_{y} w) |}^{2} . \end{array}$ For $I_{3}$ , we get $\begin{array}{rcl} I_{3} & = & C \int_{R^{N}} | v |^{\frac{4}{N - 2}} θ_{y} | w | | A^{σ} (θ_{y} w) | \\ ⩽ & C {(\int_{B (y, 1)} | v |^{\frac{2 N}{N - 2}})}^{\frac{2}{N}} {(\int_{R^{N}} | θ_{y} w |^{\frac{2 N}{N - 2 - 2 σ}})}^{\frac{N - 2 - 2 σ}{2 N}} {(\int_{R^{N}} {| A^{σ} (θ_{y} w) |}^{\frac{2 N}{N - 2 + 2 σ}})}^{\frac{N - 2 + 2 σ}{2 N}} \\ ⩽ & C ‖ v ‖_{H_{U}^{1}}^{\frac{4}{N - 2}} ({‖ A^{\frac{1 + σ}{2}} (θ_{y} w) ‖}_{L^{2}} + ‖ θ_{y} w ‖_{L^{2}}) ({‖ A^{\frac{1 + σ}{2}} (θ_{y} w) ‖}_{L^{2}} + {‖ A^{σ} (θ_{y} w) ‖}_{L^{2}}) \\ (5.14) & ⩽ & C_{ϱ_{1}} ‖ v ‖_{H_{U}^{1}}^{\frac{4}{N - 2}} \int_{R^{N}} {| A^{\frac{1 + σ}{2}} (θ_{y} w) |}^{2} + C_{ϱ_{1}} (1 + ‖ θ_{y} w ‖_{L^{2}}^{2}) . \end{array}$ Note that Theorem 4.1 and Lemma 4.2, we can take T large enough such that $\begin{array}{rcl} (5.15) & ‖ v ‖_{H_{U}^{1}}^{\frac{4}{N - 2}} ⩽ \frac{1}{8 C_{ϱ_{1}}}, for all t ⩾ T . \end{array}$ For $I_{4}$ , by (4.25), we infer $\begin{array}{rcl} I_{4} & = & ⟨ f_{1} (v), θ_{y} A^{σ} (θ_{y} w_{t}) ⟩ \\ ⩽ & C \int_{R^{N}} (1 + | v |^{γ}) θ_{y} | A^{σ} (θ_{y} w) | \\ ⩽ & C {(1 + \int_{B (y, 1)} θ_{y} | v |^{\frac{2 N γ}{N + 2 - 2 σ}})}^{\frac{N + 2 - 2 σ}{2 N}} {‖ A^{σ} (θ_{y} w) ‖}_{L^{\frac{2 N}{N - 2 + 2 σ}}} \\ ⩽ & C (1 + ‖ v ‖_{H_{U}^{1}}^{γ}) ({‖ A^{\frac{1 + σ}{2}} (θ_{y} w) ‖}_{L^{2}} + {‖ A^{σ} (θ_{y} w) ‖}_{L^{2}}) \\ (5.16) & ⩽ & C_{ϱ_{1}} (1 + ‖ θ_{y} ‖_{L^{2}}^{2}) + \frac{1}{8} \int_{R^{N}} {| A^{\frac{1 + σ}{2}} (θ_{y} w) |}^{2} . \end{array}$ For the forcing term, we have $\begin{array}{rcl} (5.17) & ⟨ g, θ_{y} A^{σ} (θ_{y} w) ⟩ ⩽ C ‖ g ‖_{L_{U}^{2}} {(\int_{R^{N}} {| A^{σ} (θ_{y} \nabla w) |}^{2})}^{\frac{1}{2}} . \end{array}$ Therefore, we get that $\begin{matrix} (5.18) & \frac{d}{d t} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} + (1 - C_{ϱ_{1}} ‖ v_{1} ‖_{H_{U}^{1}}^{2}) \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w) |}^{2} ⩽ C_{K_{ϵ}, ϱ_{1}, ‖ g ‖_{L_{lu}^{2}}} . \end{matrix}$ Applying the Gronwall lemma and integrating over $[1 + T, t]$ , we infer $\begin{array}{rcl} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w (t)) |}^{2} & ⩽ & e^{- \int_{T + 1}^{t} (1 - C_{ϱ_{1}} ‖ v_{1} (s) ‖_{H_{U}^{1}}^{2}) d s} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w (T + 1)) |}^{2} \\ (5.19) & + C_{K_{ϵ}, ϱ_{1}, ‖ g ‖_{L_{lu}^{2}}} \int_{T + 1}^{t} e^{\int_{t}^{s} (1 - C_{ϱ_{1}} ‖ v_{1} (τ) ‖_{H_{U}^{1}}^{2}) d τ} d s . \end{array}$ According to Lemma 5.4, for every $t ⩾ s ⩾ 0$ , $\begin{matrix} \int_{s}^{t} {‖ v_{1} (r) ‖}_{H_{U}^{1}}^{2} d r ⩽ ϵ (t - s) + C_{ϵ} . \end{matrix}$ Now, we choose $ϵ < \frac{1}{2 C_{ϱ_{1}}}$ , we have $\begin{array}{l} C_{K_{ϵ}, ϱ_{1}, ‖ g ‖_{L_{lu}^{2}}} \int_{T + 1}^{t} e^{\int_{t}^{s} (1 - C_{ϱ_{1}} ‖ v_{1} (τ) ‖_{H_{U}^{1}}^{2}) d τ} d s \\ ⩽ C_{K_{ϵ}, ϱ_{1}, ‖ g ‖_{L_{lu}^{2}}} e^{C_{ϱ_{1}} C_{ϵ}} \int_{T + 1}^{t} e^{(1 - C_{ϱ_{1}} ϵ) (s - t)} d s \\ ⩽ C_{K_{ϵ}, ϱ_{1}, ‖ g ‖_{L_{lu}^{2}}} \int_{T + 1}^{t} e^{\frac{s - t}{2}} d s \\ (5.20) & ⩽ C_{K_{ϵ}, ϱ_{1}, ‖ g ‖_{L_{lu}^{2}}}, \end{array}$ and $\begin{array}{l} e^{- \int_{T + 1}^{t} (1 - C_{ϱ_{1}} ‖ v_{1} (s) ‖_{H_{U}^{1}}^{2}) d s} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w (T + 1)) |}^{2} \\ (5.21) & ⩽ e^{- \frac{1}{2} (t - T - 1)} e^{C_{ϱ_{1}} C_{ϵ}} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w (T + 1)) |}^{2} . \end{array}$ From (5.19)–(5.21), we get that, for $t ⩾ T + 1$ , $\begin{array}{rcl} (5.22) & \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w (t)) |}^{2} & ⩽ & e^{- \frac{1}{2} (t - T - 1)} e^{C_{ϱ_{1}} C_{ϵ}} \int_{R^{N}} {| A^{\frac{σ + 1}{2}} (θ_{y} w (T + 1)) |}^{2} + C_{K_{ϵ}, ϱ_{1}, ‖ g ‖_{L_{lu}^{2}}} . \end{array}$ Note that T is fixed, and using Lemma 4.3, we complete the proof. □

Lemma 5.3.

Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ . Assume $B_{σ}$ is an arbitrary bounded set in $H_{lu}^{1 + σ}$ . Then there exists a constant $M_{σ} (> 0)$ which depends only on the $H_{lu}^{1 + σ}$ -bound of $B_{σ}$ such that $\begin{matrix} (5.23) & {‖ S (t) u_{0} ‖}_{H_{lu}^{1 + σ}}^{2} ⩽ M_{σ}, for all t ⩾ 0 and u_{0} \in B_{σ} . \end{matrix}$

Proof.

Multiplying (1.1) by $θ_{y} A^{σ} (θ_{y} u)$ , we infer $\begin{array}{l} ⟨ u_{t}, θ_{y} A^{σ} (θ_{y} u) ⟩ - ⟨ Δ u_{t}, θ_{y} A^{σ} (θ_{y} u) ⟩ - ⟨ Δ u, θ_{y} A^{σ} (θ_{y} u) ⟩ + ⟨ f (u), θ_{y} A^{σ} (θ_{y} u) ⟩ \\ (5.24) & = ⟨ g (x), θ_{y} A^{σ} (θ_{y} u) ⟩ . \end{array}$ The proof is similar to the proof of Lemma 5.2, so we omit it here. □

Lemma 5.4.

Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ and $σ ⩽ θ ⩽ 1$ . Then for any bounded $B_{θ} \subset H_{lu}^{1 + θ}$ . Then there exists a constant $M_{θ} (> 0)$ which depends only on the $H_{lu}^{1 + θ}$ -bound of $B_{θ}$ such that $\begin{matrix} (5.25) & {‖ S (t) (t) u_{0} ‖}_{H_{lu}^{1 + θ}}^{2} ⩽ M_{θ}, for all t ⩾ 0 and u_{0} \in B_{θ} . \end{matrix}$

Lemma 5.5.

Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ and $θ \in [σ, 1 - min {σ, \frac{4 σ}{n - 2}}]$ , and assume that the initial data set $B_{θ}$ is bounded in $H_{lu}^{1 + θ}$ , then the decomposed ingredient $w (t)$ satisfies that $\begin{matrix} (5.26) & {‖ K (t) u_{0} ‖}_{H_{lu}^{1 + θ + s_{0}}}^{2} ⩽ J_{θ}, for all t ⩾ 0 and u_{0} \in B_{θ}, \end{matrix}$ where $s_{0} = min {σ, \frac{4 σ}{n - 2}}$ and the constant $J_{θ} (> 0)$ which depends only on the $H_{lu}^{1 + θ}$ -bound of $B_{θ}$ .

Now, we state the following asymptotic regularity results:

Theorem 5.2.

Assume that f satisfies ( 1.3 )–( 1.5 ), $g (x) \in L_{lu}^{2} (R^{N})$ , and the semigroup ${S (t)}_{t ⩾ 0}$ generated by the weak solutions of ( 1.1 )–( 1.2 ) with the initial data $u_{0} \in H_{lu}^{1} (R^{N})$ . Then, there exists a set $B \subset H_{lu}^{2} (R^{N})$ (closed and bounded in $H_{lu}^{2} (R^{N})$ ), a positive constant ν and a monotonically increasing function $Q (\cdot)$ such that: for any bounded set $B \subset H_{lu}^{1} (R^{N})$ , the following estimate holds: $\begin{matrix} {dist}_{*} (S (t) B, B) ⩽ Q (‖ B ‖_{H_{lu}^{1} (R^{N})}) e^{- ν t}, \end{matrix}$ where ${dist}_{*}$ denotes the usual Hausdorff semidistance in $H_{lu}^{1} (R^{N})$ .

Proof.

Note that $\begin{matrix} B = ⋃_{t ⩾ T_{B_{0}}} S (t) B_{0}, \end{matrix}$ according to Lemma 5.4 and Lemma 4.2, we know that there is a set $A_{σ}$ which is bounded in $H_{lu}^{1 + σ} (R^{N})$ such that $\begin{matrix} {dist}_{H_{lu}^{1}} (S (t) B, A_{σ}) ⩽ {dist}_{H_{lu}^{1}} (D (t) B, A_{σ}) ⩽ Q_{1} (‖ B ‖_{H_{lu}^{1}}) e^{- k_{0} t} . \end{matrix}$ Applying Lemma 5.5 and Lemma 4.2 to $A_{σ}$ , we see that there is a set $A_{σ + s}$ which is bounded in $H_{lu}^{1 + σ + s} (R^{N})$ , such that $\begin{matrix} {dist}_{H_{lu}^{1}} (S (t) B, A_{σ + s}) ⩽ {dist}_{H_{lu}^{1}} (D (t) B, A_{σ + s}) ⩽ Q_{1} (‖ B ‖_{H_{lu}^{1}}) e^{- k_{0} t}, \end{matrix}$ where $k_{0}$ depends only on the $H_{lu}^{1}$ -bound of $A_{σ}$ . Combing this with Remark 5.1, we know that the condition in Lemma 2.3 are all satisfied. Hence we have $\begin{matrix} (5.27) & {dist}_{H_{lu}^{1}} (S (t) B, A_{σ + s}) ⩽ C Q_{1} (‖ B ‖_{H_{lu}^{1}}) e^{- k_{0} t}, \end{matrix}$ for two appropriate constants C and $k_{0}$ .

Note that $σ = min {\frac{1}{4}, \frac{N + 2 - (N - 2) γ}{2}}$ and $s_{0} = min {σ, \frac{4 σ}{n - 2}}$ are fixed, by finite steps (e.g., at most by $[\frac{1}{s}] + 2$ steps) we can infer that there is a bounded (in $H_{lu}^{2} (R^{N})$ ) set $B_{1} \subset H_{lu}^{2} (R^{N})$ such that $\begin{matrix} {dist}_{H_{lu}^{1}} (S (t) B, B_{1}) ⩽ Q_{1} (‖ B ‖_{H_{lu}^{1}}) e^{- ν t} . \end{matrix}$ Now, for any bounded set B, by Theorem 4.1, we see that there exist a T such that $\begin{matrix} S (t) B \subset B, for all t ⩾ T . \end{matrix}$ Hence, $\begin{matrix} (5.28) & {dist}_{H_{lu}^{1}} (S (t) B, B_{1}) ⩽ M e^{ν T} e^{- ν t}, \end{matrix}$ where $M = sup {‖ S (t) B ‖_{H_{lu}^{1}}, 0 ⩽ t ⩽ T} < \infty$ .

Now, we apply the attraction transitivity lemma, i.e., Lemma 2.3, again to (5.27) and (5.28), this complete the proof. □

Remark 5.3.

The $(H_{lu}^{1} (R^{N}), H_{ρ}^{1} (R^{N}))$ -global attractor given in Theorem 4.3 is bounded in the locally uniform space $H_{lu}^{2} (R^{N})$ , which appears to be optimal.

Remark 5.4.

There exists a bounded (in ( $H_{lu}^{2} (R^{N})$ )) subset which attracts exponentially every initial $H_{lu}^{1} (R^{N})$ -bounded set with respect to the $H_{lu}^{1} (R^{N})$ -norm.

Remark 5.5.

To our best knowledge, this is first time to obtain the regularity for Eqs (1.1)–(1.2) with critical nonlinearity on unbounded domain. Maybe it is a basis for further considering the asymptotic behavior, e.g., based on this result, whether the exponential attractors exist for Eqs (1.1)–(1.2) with critical nonlinearity on unbounded domain is still open.

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Attractors and asymptotic regularity for nonclassical diffusion equations in locally uniform spaces with critical exponent

Abstract

Keywords

1. Introduction

2. Preliminaries

Lemma 2.1 ([5]).

Lemma 2.2 ([5]).

Lemma 2.3 ([15]).

3. Well-posedness

Step 2 (Global existence).

Remark 3.3. Theorem 3.1 implies that the solution of Eqs (1.1)–(1.2) generates a C 0 semigroup { S ( t ) } t ⩾ 0 in the space H lu 1 ( R N ) . 4. Global attractor

4.1. Dissipation estimates

References

Remark 3.3.
Theorem 3.1 implies that the solution of Eqs (1.1)–(1.2) generates a $C^{0}$ semigroup ${S (t)}_{t ⩾ 0}$ in the space $H_{lu}^{1} (R^{N})$ .

4. Global attractor