Pullback attractors for a class of non-autonomous reaction-diffusion equations in R
           n
          
         
        
        $\mathbb{R}^{n}$

Qiangheng Zhang

doi:10.1186/s13661-017-0879-5

Research
Open Access

Pullback attractors for a class of non-autonomous reaction-diffusion equations in $\mathbb{R}^{n}$

Qiangheng Zhang¹Email author

Boundary Value Problems20172017:146

https://doi.org/10.1186/s13661-017-0879-5

Received: 7 August 2017

Accepted: 25 September 2017

Published: 6 October 2017

Abstract

The aim of this paper is to consider the dynamical behaviour for a class of non-autonomous reaction-diffusion equations in $\mathbb{R}^{n}$, where the external force $g(x,t)$ satisfies only a certain integrability condition. The existence of $(L^{2}(\mathbb{R}^{n}),L^{2}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback attractors and $(L^{2}(\mathbb{R}^{n}),L^{p}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback attractors is obtained for this evolution equation.

Keywords

pullback attractorsdynamical behaviournon-autonomous equations

MSC

35B4135B4535K15

1 Introduction

In this paper, we consider the asymptotic behaviour of solutions for the following non-autonomous reaction-diffusion equations defined in the whole space:

$$\begin{aligned} \textstyle\begin{cases} u_{t}-\nu\Delta u+\lambda u+f_{1}(u)+a(x)f_{2}(u)=g(x, t), & \text{in } \mathbb{R}^{n}\times[\tau, \infty), \\ u(x,\tau)=u_{\tau}, &\text{in } \mathbb{R}^{n}, \end{cases}\displaystyle \end{aligned}$$

(1.1)

where ν and λ are positive constants. Assume that nonlinear terms $f_{1}(u), f_{2}(u) \in C^{1}(\mathbb{R}; \mathbb{R})$ satisfy the following conditions:

$$\begin{aligned} \alpha_{1}\vert u\vert ^{p}- \beta_{1}\vert u\vert ^{2}\leq f_{1}(u)u\leq \alpha_{2}\vert u\vert ^{p}+ \beta_{2}\vert u \vert ^{2} \quad \text{and} \quad f_{1}^{\prime}(u) \geq-l_{1} \end{aligned}$$

(1.2)

with $p>2$ and $\lambda>\beta_{1}$,

$$\begin{aligned} \alpha_{3}\vert u\vert ^{p}- \beta_{3}\leq f_{2}(u)u\leq\alpha_{4}\vert u \vert ^{p}+\beta _{4} \quad \text{and} \quad f_{2}^{\prime}(u)\geq-l_{2} \end{aligned}$$

(1.3)

with $p>2$, where $\alpha_{i}$, $\beta_{i}$, $i=1, 2, 3, 4$, and $l_{i}$, $i=1, 2$ are positive constants. Furthermore, $a(x)$ is a function in $\mathbb{R}^{n}$ and the external force $g(x, t) \in L^{2}_{\mathrm {loc}}(\mathbb{R}; L^{2}(\mathbb{R}^{n}))$ satisfies the following conditions:

$$\begin{aligned}& a(x)\in L^{1} \bigl(\mathbb{R}^{n} \bigr)\cap L^{\infty} \bigl(\mathbb{R}^{n} \bigr) \quad \text{and} \quad a(x)>0, \end{aligned}$$

(1.4)

$$\begin{aligned}& \int_{-\infty}^{t}e^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} ^{2}\,ds< \infty, \quad\text{for all } t\in \mathbb{R}, \sigma\in(0, \lambda-\beta_{1}). \end{aligned}$$

(1.5)

In the last decade, the autonomous and non-autonomous infinite dimensional dynamical systems have been studied extensively by many authors (see, e.g. [1–8] and the references therein). The concept of pullback attractors was proposed in [9] when the authors considered the asymptotic behaviour of random dynamical systems. Such attractors is a parameterised family $\{A(t)\}_{t\in\mathbb{R}}$ of invariant compact sets, which attract the trajectories of the systems when the initial instant of time goes to −∞ and the final time remains fixed. Later on, the pullback attractors were extended to non-autonomous dynamical systems. In the last two decades, the theory of pullback attractors has been developed for non-autonomous dynamical systems and random dynamical systems (see, e.g. [10–12] and the references therein). In [10], the authors introduced the notion of $\mathscr{D}$-pullback attractors, which requires that the process $U(t, \tau)$ associated with the systems be $\mathscr{D}$-pullback asymptotically compact.

It is well known that the Sobolev embeddings are no longer compact in unbounded domain, and so it is difficult to verify the process $U(t, \tau)$ associated with the systems to be pullback asymptotically compact. To overcome this drawback, in [13], using the idea of Wang [5], the authors proved the existence of pullback attractors in $L^{2}(\mathbb{R}^{n})$ and $H^{1}(\mathbb{R}^{n})$ for non-autonomous reaction-diffusion equations defined on $\mathbb{R} ^{n}$. Recently, motivated by [3], the authors of [14] gave a new method to prove the existence of $\mathscr{D}$-pullback attractors by using the technique of non-compactness measure, and this method only needs the process $U(t, \tau)$ associated with the systems to be norm-to-weak continuous (see Definition 2.1) in the phase space.

As we know, the solutions may be unbounded for many non-autonomous systems when time tends to infinity, and we cannot obtain the existence of a uniform attractor for these systems. So we prove the existence of a pullback attractor to overcome this drawback. In this paper, we use a different approach from the article [13] to prove the existence of pullback attractors, and we improve the model equation as Eq. (1.1), which amounts to putting a weight function partially on the nonlinearity. We can also replace the conditions for the nonlinearity $f(u)$ as given in [13] that $f(u)$ satisfies only a Sobolev growth rate with some weak assumptions. For Eq. (1.1), the $(L^{2}(\mathbb{R} ^{n}), L^{2}(\mathbb{R}^{n}))$-global attractor, $(L^{2}(\mathbb{R} ^{n}), L^{p}(\mathbb{R}^{n}))$-global attractor and $(L^{2}( \mathbb{R}^{n}), H^{1}(\mathbb{R}^{n}))$-global attractor were proved in [15, 16]. Using the new method in [14], we prove the existence of $\mathscr{D}$-pullback attractors in $L^{2}(\mathbb{R} ^{n})$ for Eq. (1.1) and, motivated by the idea in [17, 18], we obtain the existence of $\mathscr{D}$-pullback attractors in $L^{p}(\mathbb{R}^{n})$ for Eq. (1.1). This new method has been used successfully in many papers (see, e.g. [14, 17, 19, 20] and the references therein).

For convenience, the letter C denotes a constant which may be different from line to line and even in the same line. We use $\Vert \cdot \Vert $ and $(\cdot, \cdot)$ for the usual norm and the inner product of $L^{2}(\mathbb{R}^{n})$, respectively. We denote by $\Vert \cdot \Vert _{p}$ the norm of $L^{p}(\mathbb{R}^{n})$ ($1\leq p\leq\infty$) and by $\Vert \cdot \Vert _{H^{1}} $ the norm of $H^{1}(\mathbb{R}^{n})$. In general, $m(e)$ is the Lebesgue measure of $e\subset\mathbb{R} ^{n}$. $\Vert \cdot \Vert _{E}$ denotes the norm of any Banach space E and $B(E)$ is the set of all bounded subsets of E. Let $X, Y \subset E$, denote by dist $(X, Y)=\sup_{x\in X}\inf_{y\in Y}\,d(x, y)$ the semidistance between X and Y.

2 Preliminaries

In this section, we first recall the basic definitions and theorems.

Definition 2.1

[14]

Let X be a complete metric space and $\{U(t, \tau) \}=\{U(t, \tau): t\geq\tau, {\tau\in\mathbb{R}}\}$ be a two-parameter family of mappings acting on X: $U(t, \tau): X\rightarrow X$, $t\geq \tau$, $\tau\in\mathbb{R}$. We say that $\{U(t, \tau)\}_{\tau \leq t}$ is a continuous process (or norm-to-weak continuous process) in X if

(1)

$U(t, s)U(s, \tau)=U(t, \tau)$, $\forall t\geq s\geq\tau$,
(2)

$U(\tau, \tau)=\mathit{Id}$ is the identity operator, $\tau\in \mathbb{R}$,
(3)

$x\rightarrow U(t, \tau)x$ is continuous in X

(or $U(t, \tau)x_{n}\rightharpoonup U(t, \tau)x$ if $x_{n}\rightarrow x$, $\forall t\geq\tau$, $\tau\in\mathbb{R}$).

Suppose that $\mathscr{D}$ is a nonempty class of parameterised sets $\hat{\mathscr{D}}=\{D(t):t\in\mathbb{R}\}\subset B(E)$.

Definition 2.2

[14]

The process $\{U(t, \tau)\}_{\tau\leq t}$ is said to be $\mathscr{D}$-pullback asymptotically compact if, for any $t\in\mathbb{R}$ and any $\hat{\mathscr{D}}\in\mathscr{D}$, and any sequence $\tau_{n}\rightarrow-\infty$, any sequence $x_{n} \in D(\tau_{n})$, the sequence $\{U(t, \tau_{n})x_{n}\}$ is precompact in X.

Definition 2.3

[14]

It is said that $\hat{\mathscr{B}}\in\mathscr{D}$ is $\mathscr{D}$-pullback absorbing for the process $\{U(t, \tau)\}_{\tau\leq t}$ if, for any $t\in\mathbb{R}$ and any $\hat{\mathscr{D}}\in\mathscr{D}$, there exists $\tau_{0}(t, \hat{\mathscr{D}})\leq t$ such that $U(t, \tau) D(\tau)\subset B(t)$ for all $\tau\leq\tau_{0}(t, \hat{\mathscr{D}})$.

Definition 2.4

[14]

The family $\hat{A}=\{A(t): t\in\mathbb{R}\}\subset B(E)$ is said to be a $\mathscr{D}$-pullback attractor for $U(t, \tau)$ if

(1)

$A(t)$ is compact for all $t\in\mathbb{R}$,
(2)

Â is invariant, i.e.
$$ U(t, \tau)A(\tau)=A({\tau}) \quad\text{for all } t\geq\tau, $$
(3)

Â is $\mathscr{D}$-pullback attracting, i.e.
$$ \lim_{\tau\rightarrow-\infty} \text{dist} \bigl(U(t, \tau)D(\tau), A(t) \bigr)=0 \quad\text{for all }\hat{\mathscr{D}}\in\mathscr{D} \text{ and all } t\in \mathbb{R}, $$
(4)

if $\{C(t)\}_{t\in\mathbb{R}}$ is another family of closed attracting sets, then $A(t)\subset C(t)$ for all $t\in\mathbb{R}$.

Definition 2.5

[21]

Let M be a metric space and A be a bounded subset of M. The Kuratowski measure of non-compactness $\alpha(A)$ is defined by

$$\begin{aligned} \alpha(A)=\inf \{\delta>0\mid A \text{ admits a finite cover by sets of diameter}\leq\delta \}. \end{aligned}$$

It has the following properties.

Lemma 2.1

[21]

Let $B, B_{1}, B_{2}\in B(E)$. Then

(1)

$\alpha(B)=0\Leftrightarrow\alpha(N(B, \varepsilon))\leq2 \varepsilon\Leftrightarrow\bar{B}$ is compact;
(2)

$\alpha(B_{1}+B_{2})\leq\alpha(B_{1})+\alpha(B_{2})$;
(3)

$\alpha(B_{1})\leq\alpha(B_{2})$ whenever $B_{1}\subset B_{2}$;
(4)

$\alpha(B_{1}\cup B_{2})\leq\max\{\alpha(B_{1}), \alpha(B _{2})\}$;
(5)

$\alpha(B)=\alpha(\bar{B})$;
(6)

if B is a ball of radius ε, then $\alpha(B)\leq2 \varepsilon$.

Definition 2.6

[14]

A process $\{U(t, \tau)\}_{\tau\leq t}$ is called $\mathscr{D}$-pullback ω-limit compact if for any $\varepsilon>0$ and $\hat{\mathscr{D}}\in\mathscr{D}$, there exists $\tau_{0}(t, \hat{\mathscr{D}})\leq t$ such that $\alpha ( \bigcup_{ \tau\leq\tau_{0}}U(t, \tau)D(\tau) ) \leq\varepsilon$.

Theorem 2.1

[14]

Let $\{U(t, \tau)\}_{\tau\leq t}$ be a process on X. Then $\{U(t, \tau)\}_{\tau\leq t}$ is $\mathscr{D}$-pullback asymptotically compact if and only if $\{U(t, \tau)\}_{\tau\leq t}$ is $\mathscr{D}$-pullback ω-limit compact.

Theorem 2.2

[14]

Let $\{U(t, \tau)\}_{\tau\leq t}$ be a norm-to-weak continuous process such that $\{U(t, \tau)\}_{\tau\leq t}$ is $\mathscr{D}$-pullback ω-limit compact. If there exists a family of $\mathscr{D}$-pullback absorbing sets $\{B(t): t \in\mathbb{R}\}\in\mathscr{D}$, i.e. for any $t\in\mathbb{R}$ and $\hat{\mathscr{D}}\in\mathscr{D}$, there exists $\tau_{0}(t, \hat{\mathscr{D}})\leq t$ such that $U(t, \tau)D(\tau)\subset B(t)$ for all $\tau\leq\tau_{0}$, then there exists a $\mathscr{D}$-pullback attractor $\mathcal{A}=\{A(t):t\in\mathbb{R} \}$ and

$$\begin{aligned} A(t)=\omega(\hat{B}, t)=\bigcap_{s\leq t}\overline{ \bigcup_{\tau \leq s}U(t, \tau)B(\tau)}. \end{aligned}$$

Remark

Obviously, a continuous process and a weak continuous process are both norm-to-weak continuous processes.

Theorem 2.3

[17]

Let Ω be a domain in $\mathbb{R}^{n}$, $\{U(t, \tau)\}_{\tau\leq t}$ be a process on $L^{p}(\Omega)$ and $L^{q}(\Omega)$ ($p>q\geq1$) and $\{U(t, \tau)\}_{\tau\leq t}$ satisfy the following two assumptions:

(1)

$\{U(t, \tau)\}_{\tau\leq t}$ is $\mathscr{D}$-pullback ω-limit compact in $L^{q}(\Omega)$;
(2)

for any $\varepsilon>0$, $\hat{\mathscr{B}}\in\mathscr{D}$, there exist $M(\varepsilon, \hat{\mathscr{B}})$ and $\tau_{1}=\tau_{1}( \varepsilon, \hat{\mathscr{B}})\leq t$ such that
$$\begin{aligned} \biggl( \int_{\Omega(\vert U(t, \tau)\vert \geq M)} \bigl\vert U(t, \tau)u_{\tau } \bigr\vert ^{p}\,dx \biggr) ^{\frac{1}{p}}< 2^{-\frac{2p+2}{p}}\varepsilon \quad \textit{for any } u_{\tau}\in B(\tau) \textit{ and } \tau\geq \tau_{1}. \end{aligned}$$

Then $\{U(t, \tau)\}_{\tau\leq t}$ is $\mathscr{D}$-pullback ω-limit compact in $L^{p}(\Omega)$.

Theorem 2.4

[13]

Let X, Y be two Banach spaces with the norms $\Vert \cdot \Vert _{X}$ and $\Vert \cdot \Vert _{Y}$, respectively. Let $\{U(t, \tau)\}_{\tau\leq t}$ be a continuous process on X and a process on Y. Assume that the family $\hat{\mathscr{B}}_{0}=\{B_{0}(t): t \in\mathbb{R}\}$ is $(X, X)$-$\mathscr{D}$-pullback absorbing for $U(t, \tau)$, and for any $t\in\mathbb{R}$ and any sequence $\tau_{n}\rightarrow-\infty$, any sequence $x_{n}\in B_{0}( \tau_{n})$, the sequence $\{U(t, \tau_{n})x_{n}\}$ is precompact in X. Then the family of sets $\mathcal{A}=\{A(t):t\in\mathbb{R}\}$, where

$$\begin{aligned} A(t)=\bigcap_{s\leq t}\overline{\bigcup _{\tau\leq s}U(t, \tau)B( \tau)}^{X} \end{aligned}$$

is a $(X, X)$-$\mathscr{D}$-pullback attractor for $\{U(t, \tau)\}_{\tau\leq t}$, where $\overline{A}^{X}$ denotes the closure of A with respect to the norm topology in X.

Furthermore, if the family $\hat{\mathscr{B}}_{1}=\{B_{1}(t): t\in \mathbb{R}\}$ is $(X, Y)$-$\mathscr{D}$-pullback absorbing for $\{U(t, \tau)\}_{\tau\leq t}$, and it satisfies that, for any $t\in\mathbb{R}$ and any sequence $\tau_{n}\rightarrow- \infty$, any sequence $x_{n}\in B_{1}(\tau_{n})$, the sequence $\{U(t, \tau_{n})x_{n}\}$ is precompact in Y. Then the family of sets $\mathcal{A}^{\prime}=\{A^{\prime}(t):t\in\mathbb{R}\}$, where

$$\begin{aligned} A^{\prime}(t)=\bigcap_{s\leq t}\overline{\bigcup _{\tau\leq s}U(t, \tau) \bigl( B_{0}(\tau)\cap B_{1}(\tau) \bigr) }^{X}=\bigcap _{s \leq t}\overline{\bigcup_{\tau\leq s}U(t, \tau) \bigl( B_{0}(\tau) \cap B_{1}(\tau) \bigr) }^{Y} \end{aligned}$$

is a $(X, Y)$-$\mathscr{D}$-pullback attractors for $\{U(t, \tau)\}_{\tau\leq t}$.

Remark

When $\{U(t, \tau)\}_{\tau\leq t}$ is only a process on Y, we also prove $\mathcal{A}^{\prime}=\{A^{\prime}(t):t\in\mathbb{R}\}$ is a $(X, Y)$-$\mathscr{D}$-pullback attractor for $\{U(t, \tau)\}_{\tau\leq t}$.

Lemma 2.2

Let $\{U(t, \tau)\}_{\tau\leq t}$ be a process on $L^{p}(\mathbb{R} ^{n})$ ($p\geq1$), $\hat{\mathscr{B}}_{1}=\{B_{1}(t): t\in \mathbb{R}\}$ is $(X, Y)$-$\mathscr{D}$-pullback absorbing for $\{U(t, \tau)\}_{\tau\leq t}$. Then, for any $\varepsilon>0$, $t\in\mathbb{R}$ and $\hat{\mathscr{D}}\in \mathscr{D}\subset B(L^{p}(\mathbb{R}^{n}))$, there exist $M(t, \varepsilon)$ and $\tau_{0}=\tau_{0}(t, \varepsilon)$ such that

$$\begin{aligned} m \bigl( \mathbb{R}^{n} \bigl( \bigl\vert U(t, \tau) \bigr\vert \geq M(t, \varepsilon) \bigr) \bigr) < \varepsilon\quad \textit{for all } u_{\tau}\in D(\tau) \textit{ and } \tau\leq\tau_{0}. \end{aligned}$$

The proof of the above lemma is identical to the proof of Lemma 5.2 in [18].

Using the standard Faedo-Galerkin method (see [6, 7]), it is easy to prove the following lemma.

Lemma 2.3

Assume that (1.2)-(1.5) hold and $g\in L ^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$. Then, for any $T>0$, $u_{\tau}\in L^{2}(\mathbb{R}^{n})$, $\tau \in\mathbb{R}$ and $T\geq\tau$, there exists a unique weak solution $u(x,t)$ for Eq. (1.1) satisfying

$$\begin{aligned} u\in C \bigl([\tau, T]; L^{2} \bigl(\mathbb{R}^{n} \bigr) \bigr) \cap L^{p} \bigl(\tau, T; L^{p} \bigl( \mathbb{R}^{n} \bigr) \bigr)\cap L^{2} \bigl(\tau, T; H^{1} \bigl( \mathbb{R}^{n} \bigr) \bigr). \end{aligned}$$

Furthermore, $u_{\tau} \mapsto u(t, \tau; u_{\tau})$ is continuous in $L^{2}(\mathbb{R}^{n})$.

Based on Lemma 2.3, we can define a continuous process $\{U(t, \tau)\}_{\tau\leq t}$ in $L^{2}(\mathbb{R}^{n})$ by

$$\begin{aligned} U(t, \tau)u_{\tau}=u(t) \quad \text{for all } t\geq\tau, \end{aligned}$$

(2.1)

where $u(t)$ is the solution of Eq. (1.1) with the initial value $u(x, \tau)=u_{\tau}\in L^{2}(\mathbb{R}^{n})$. Moreover, we also know that $\{U(t, \tau)\}_{\tau\leq t}$ is a process in $L^{p}( \mathbb{R}^{n})$.

3 Main results

3.1 $(L^{2}(\mathbb{R}^{n}),L^{2}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback attractors

Firstly, the following lemma ensures a $\mathscr{D}$-pullback absorbing set in $L^{2}( \mathbb{R}^{n})$.

Lemma 3.1

Assume that (1.2)-(1.4) hold and the external force $g\in L^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$ satisfies (1.5). Then, for any $\hat{\mathscr{D}}\in \mathscr{D}\subset B(L^{2}(\mathbb{R}^{n}))$ and any $t\in\mathbb{R}$, there exists $\tau_{0}(t, \hat{\mathscr{D}})\leq t$ such that

$$\begin{aligned} \bigl\Vert U(t, \tau)u_{\tau} \bigr\Vert \leq R_{0}(t) \quad \textit{for all } \tau\leq\tau_{0}(t, \hat{ \mathscr{D}}) \textit{ and all } u_{\tau}\in D(\tau), \end{aligned}$$

(3.1)

where $R_{0}(t)= ( \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}+\frac{2e ^{-\sigma t}}{\lambda-\beta_{1}}\int_{-\infty}^{t}e^{\sigma r}\Vert g(x, r)\Vert ^{2}\,dr ) ^{\frac{1}{2}}$.

Proof

Taking the inner product of (1.1) with u in $L^{2}( \mathbb{R}^{n})$, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert u\Vert ^{2}+\nu \Vert \nabla u\Vert ^{2}+\lambda \Vert u\Vert ^{2}+ \bigl(f_{1}(u), u \bigr)+ \bigl(a(x)f_{2}(u), u \bigr)= \bigl(g(x, t), u \bigr). \end{aligned}$$

Due to (1.2)-(1.4) and Young’s inequality, we get

$$\begin{aligned}& \frac{d}{dt}\Vert u\Vert ^{2}+(\lambda- \beta_{1})\Vert u\Vert ^{2}\leq2\beta_{3} \bigl\Vert a(x) \bigr\Vert _{1}+\frac{\Vert g(x, t)\Vert ^{2}}{\lambda-\beta_{1}}, \end{aligned}$$

(3.2)

$$\begin{aligned}& \begin{aligned}[b] &\frac{d}{dt}\Vert u\Vert ^{2}+(\lambda-\beta_{1})\Vert u\Vert ^{2}+2\nu \Vert \nabla u \Vert ^{2}+2\alpha_{1}\Vert u\Vert _{p}^{p}+2\alpha_{3} \int_{\mathbb{R}^{n}}a(x)\vert u\vert ^{p}\,dx \\ &\quad\leq2\beta_{3} \bigl\Vert a(x) \bigr\Vert _{1}+ \frac{\Vert g(x, t)\Vert ^{2}}{\lambda-\beta _{1}}. \end{aligned} \end{aligned}$$

(3.3)

By (3.2), we obtain

$$\begin{aligned} \frac{d}{dr} \bigl(e^{\sigma r}\Vert u\Vert ^{2} \bigr)+( \lambda-\beta_{1}-\sigma)e^{ \sigma r}\Vert u\Vert ^{2} \leq2\beta_{3} \bigl\Vert a(x) \bigr\Vert _{1}e^{\sigma r}+ \frac{\Vert g(x, r)\Vert ^{2}}{\lambda-\beta_{1}}e^{\sigma r}. \end{aligned}$$

Integrating over the interval $[\tau, t]$ and noting that $\sigma \in(0, \lambda-\beta_{1})$, we have

$$\begin{aligned} e^{\sigma t} \bigl\Vert u(t) \bigr\Vert ^{2} \leq& \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma }e^{\sigma t}+\frac{1}{\lambda-\beta_{1}} \int_{\tau}^{t}e^{\sigma r} \bigl\Vert g(x, r) \bigr\Vert ^{2}\,dr+e^{\sigma\tau} \Vert u_{\tau} \Vert ^{2} \\ \leq& \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}e^{\sigma t}+\frac{1}{ \lambda-\beta_{1}} \int_{-\infty}^{t}e^{\sigma r} \bigl\Vert g(x, r) \bigr\Vert ^{2}\,dr+e ^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}. \end{aligned}$$

(3.4)

Thus, we get

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert ^{2}\leq\frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}+e^{-\sigma t}e ^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}+\frac{e^{-\sigma t}}{\lambda-\beta _{1}} \int_{-\infty}^{t}e^{\sigma r} \bigl\Vert g(x, r) \bigr\Vert ^{2}\,dr, \end{aligned}$$

and this implies (3.1). □

Let $\hat{\mathscr{B}}_{0}=\{B_{0}(t): t\in\mathbb{R}\}$, where

$$\begin{aligned} B_{0}(t)= \bigl\{ u\in L^{2} \bigl( \mathbb{R}^{n} \bigr): \Vert u\Vert \leq R_{0}(t) \bigr\} . \end{aligned}$$

(3.5)

By Lemma 3.1, it is easy to know that the family $\hat{\mathscr{B}} _{0}$ is $(L^{2}(\mathbb{R}^{n}),L^{2}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback absorbing for the process $\{U(t, \tau)\}_{ \tau\leq t}$ defined by (2.1) and

$$\begin{aligned} e^{\sigma t} \bigl( R_{0}(t) \bigr) ^{2} \rightarrow0 \quad \text{as } t\rightarrow-\infty. \end{aligned}$$

(3.6)

Let $F_{1}(u)=\int_{0}^{u}f_{1}(s)\,ds$ and $F_{2}(u)=\int_{0}^{u}f_{2}(s)\,ds$. By (1.2)-(1.3), there exist positive constants $\tilde{\alpha_{i}}$, $\tilde{\beta_{i}}$, $i=1, 2, 3, 4$, such that

$$\begin{aligned}& \tilde{\alpha}_{1}\vert u\vert ^{p}-\tilde{ \beta}_{1}\vert u\vert ^{2}\leq F_{1}(u) \leq \tilde{\alpha}_{2}\vert u\vert ^{p}+\tilde{ \beta}_{2}\vert u\vert ^{2}, \quad \quad \lambda>2 \tilde{ \beta}_{1}, \end{aligned}$$

(3.7)

$$\begin{aligned}& \tilde{\alpha}_{3}\vert u\vert ^{p}-\tilde{ \beta}_{3}\leq F_{2}(u)\leq \tilde{\alpha}_{4} \vert u\vert ^{p}+\tilde{\beta}_{4}. \end{aligned}$$

(3.8)

Lemma 3.2

Assume that (1.2)-(1.4) hold and the external force $g\in L^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$ satisfies (1.5). Then, for any $\hat{\mathscr{D}}\in \mathscr{D}\subset B(L^{2}(\mathbb{R}^{n}))$ and any $t\in\mathbb{R}$, there exists $\tau_{1}(t, \hat{\mathscr{D}})\leq t$ such that

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert ^{2}+ \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+ \bigl\Vert u(t) \bigr\Vert ^{p}_{p}\leq \bigl( R_{1}(t) \bigr) ^{2} \quad \textit{for all } \tau\leq\tau_{1}(t, \hat{\mathscr{D}}) \textit{ and all } u_{\tau}\in D(\tau), \end{aligned}$$

(3.9)

where $R_{1}(t)=C ( \frac{\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}+\frac{e ^{-\sigma t}}{\lambda-\beta_{1}}\int_{-\infty}^{t}e^{\sigma r}\Vert g(x, r)\Vert ^{2}\,dr ) ^{\frac{1}{2}}$ and the positive constant C is independent of t and $\hat{\mathscr{D}}$.

Proof

Multiplying (3.3) by $e^{\sigma t}$, we have

$$\begin{aligned}& \frac{d}{dt} \bigl(e^{\sigma t} \bigl\Vert u(t) \bigr\Vert ^{2} \bigr)+(\lambda-\beta_{1}-\sigma)e ^{\sigma t} \bigl\Vert u(t) \bigr\Vert ^{2}+2\nu e^{\sigma t} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & \quad\quad{} +2\alpha_{1}e^{\sigma t}\Vert u\Vert _{p}^{p}+2\alpha _{3}e^{\sigma t} \int_{\mathbb{R}^{n}}a(x)\vert u\vert ^{p}\,dx \\ & \quad \leq2\beta_{3}e^{\sigma t} \bigl\Vert a(x) \bigr\Vert _{1}+e^{\sigma t}\frac{\Vert g(x, t) \Vert ^{2}}{\lambda-\beta_{1}}. \end{aligned}$$

Let $\tau< t-1$ and $r\in[\tau, t-1]$, integrating over the interval $[r, r+1]$, we get

$$\begin{aligned}& e^{\sigma(r+1)} \bigl\Vert u(r+1) \bigr\Vert ^{2}+(\lambda- \beta_{1}-\sigma) \int_{r} ^{r+1}e^{\sigma s} \bigl\Vert u(s) \bigr\Vert ^{2}\,ds+2\nu \int_{r}^{r+1}e^{\sigma s} \bigl\Vert \nabla u(s) \bigr\Vert ^{2}\,ds \\& \quad\quad{} +2\alpha_{1} \int_{r}^{r+1}e^{\sigma s} \bigl\Vert u(s) \bigr\Vert _{p}^{p}\,ds+2\alpha _{3} \int_{r}^{r+1}e^{\sigma s} \int_{\mathbb{R}^{n}}a(x) \bigl\vert u(s) \bigr\vert ^{p}\,dx \,ds \\& \quad \leq2\beta_{3} \bigl\Vert a(x) \bigr\Vert _{1} \int_{r}^{r+1}e^{\sigma s}\,ds+ \int_{r} ^{r+1}e^{\sigma s}\frac{\Vert g(x, s)\Vert ^{2}}{\lambda-\beta_{1}} \,ds+e^{ \sigma r} \bigl\Vert u(r) \bigr\Vert ^{2}. \end{aligned}$$

By (3.4), we find

$$\begin{aligned}& \int_{r}^{r+1}e^{\sigma s} \biggl( \bigl\Vert u(s) \bigr\Vert ^{2}+ \bigl\Vert \nabla u(s) \bigr\Vert ^{2}+ \bigl\Vert u(s) \bigr\Vert _{p}^{p}+ \int_{\mathbb{R}^{n}}a(x) \bigl\vert u(s) \bigr\vert ^{p}\,dx \biggr) \,ds \\& \quad \leq C \biggl( \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}e^{\sigma (r+1)}+\frac{1}{ \lambda-\beta_{1}} \int_{\tau}^{r+1}e^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds+e ^{\sigma\tau} \Vert u_{\tau} \Vert ^{2} \biggr) \\& \quad \leq C \biggl( \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}e^{\sigma t}+e ^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}+\frac{1}{\lambda-\beta_{1}} \int_{-\infty}^{t}e^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds \biggr) . \end{aligned}$$

Thus, by (3.7) and (3.8), we can obtain

$$\begin{aligned}& \int_{r}^{r+1}e^{\sigma s} \biggl( \frac{\nu}{2}\Vert \nabla u\Vert ^{2}+\frac{ \lambda}{2}\Vert u\Vert ^{2}+ \int_{\mathbb{R}^{n}}F_{1}(u)\,dx+ \int_{\mathbb{R}^{n}}a(x)F_{2}(u)\,dx \biggr) \,ds \\& \quad \leq C \biggl( \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}e^{\sigma t}+e ^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}+\frac{1}{\lambda-\beta_{1}} \int_{-\infty}^{t}e^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds \biggr) . \end{aligned}$$

(3.10)

Multiplying (1.1) by $u_{t}$ and integrating on $\mathbb{R} ^{n}$, we have

$$\begin{aligned}& \Vert u_{t}\Vert ^{2}+\frac{d}{dt} \biggl( \frac{\nu}{2}\Vert \nabla u\Vert ^{2}+\frac{ \lambda}{2}\Vert u\Vert ^{2}+ \int_{\mathbb{R}^{n}}F_{1}(u)\,dx+ \int_{\mathbb{R}^{n}}a(x)F_{2}(u)\,dx \biggr) \\& \quad \leq\frac{1}{2} \bigl( \bigl\Vert g(x, t) \bigr\Vert ^{2}+\Vert u_{t}\Vert ^{2} \bigr). \end{aligned}$$

And then

$$\begin{aligned} \frac{d}{dt} \biggl( \frac{\nu}{2}\Vert \nabla u \Vert ^{2}+\frac{\lambda}{2}\Vert u \Vert ^{2}+ \int_{\mathbb{R}^{n}}F_{1}(u)\,dx+ \int_{\mathbb {R}^{n}}a(x)F_{2}(u)\,dx \biggr) \leq \frac{1}{2} \bigl\Vert g(x, t) \bigr\Vert ^{2}. \end{aligned}$$

(3.11)

It follows from (3.11) that

$$\begin{aligned}& \frac{d}{dr}e^{\sigma r} \biggl( \frac{\nu}{2}\Vert \nabla u \Vert ^{2}+\frac{ \lambda}{2}\Vert u\Vert ^{2}+ \int_{\mathbb{R}^{n}}F_{1}(u)\,dx+ \int_{\mathbb{R}^{n}}a(x)F_{2}(u)\,dx \biggr) \\& \quad \leq\sigma e^{\sigma r} \biggl( \frac{\nu}{2}\Vert \nabla u \Vert ^{2}+\frac{ \lambda}{2}\Vert u\Vert ^{2}+ \int_{\mathbb{R}^{n}}F_{1}(u)\,dx+ \int_{\mathbb{R}^{n}}a(x)F_{2}(u)\,dx \biggr) +\frac{e^{\sigma r}}{2} \bigl\Vert g(x,t) \bigr\Vert ^{2}. \end{aligned}$$

By (1.5), (3.7), (3.8), (3.10) and the uniform Gronwall inequality, we obtain

$$\begin{aligned}& \frac{\nu}{2}\Vert \nabla u\Vert ^{2}+ \frac{\lambda}{2}\Vert u\Vert ^{2}+ \int_{\mathbb{R}^{n}}F_{1}(u)\,dx+ \int_{\mathbb{R}^{n}}a(x)F_{2}(u)\,dx \\& \quad\leq C \biggl( \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}+e^{-\sigma(t- \tau)}\Vert u_{\tau} \Vert ^{2}+\frac{e^{-\sigma t}}{\lambda-\beta_{1}} \int_{-\infty}^{t}e^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds \biggr) . \end{aligned}$$

(3.12)

It follows from (3.7) and (3.8) that

$$\begin{aligned}& \bigl\Vert u(t) \bigr\Vert ^{2}+ \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+ \bigl\Vert u(t) \bigr\Vert ^{p}_{p} \\& \quad\leq C \biggl( \frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}+e^{-\sigma(t- \tau)}\Vert u_{\tau} \Vert ^{2}+\frac{e^{-\sigma t}}{\lambda-\beta_{1}} \int_{-\infty}^{t}e^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds \biggr) , \end{aligned}$$

and this implies (3.9). □

Lemma 3.3

Assume that (1.2)-(1.4) hold and the external force $g\in L^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$ satisfies (1.5). Let the family $\hat{\mathscr{B}}_{0}=\{B _{0}(t): t\in\mathbb{R}\}$ be defined by (3.5). Then, for any $\varepsilon\geq0$ and any $t\in\mathbb{R}$, there exist $\tilde{k}=\tilde{k}(t, \varepsilon)>0$ and $\tau^{\prime}(t, \varepsilon)$ such that

$$\begin{aligned} \int_{\vert x\vert \geq k} \bigl\vert U(t, \tau)u_{\tau} \bigr\vert ^{2}\,dx\leq \varepsilon \quad \textit{for all } k\geq\tilde{k}, \tau\leq \tau^{\prime}(t, \varepsilon ) \textit{ and } u_{\tau}\in B_{0}(\tau). \end{aligned}$$

(3.13)

Proof

Choose a smooth function θ such that $0\leq\theta(s)\leq1$ for $s\in\mathbb{R}^{+}$,

$$\begin{aligned} \theta(s)= \textstyle\begin{cases} 0, &0\leq s\leq1, \\ 1, &s\geq2, \end{cases}\displaystyle \end{aligned}$$

and there exists a constant c such that $\vert \theta^{\prime}(s)\vert \leq c$.

Multiplying (1.1) by $\theta^{2}(\frac{\vert x\vert ^{2}}{k^{2}})u$ and integrating on $\mathbb{R}^{n}$, we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k ^{2}} \biggr) \vert u\vert ^{2}\,dx-\nu \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k ^{2}} \biggr)u \Delta u\,dx+\lambda \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k ^{2}} \biggr) \vert u\vert ^{2}\,dx \\& \quad=- \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)f_{1}(u)u\,dx- \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)a(x)f_{2}(u)u\,dx+ \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)g(x, t)u\,dx \\& \quad\leq\beta_{1} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac {\vert x\vert ^{2}}{k^{2}} \biggr) \vert u\vert ^{2}\,dx- \alpha_{1} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \vert u\vert ^{p}\,dx +\beta_{3} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)a(x)\,dx \\& \quad\quad{} -\alpha_{3} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac {\vert x\vert ^{2}}{k^{2}} \biggr)a(x)\vert u\vert ^{p}\,dx+ \frac{\lambda-\beta_{1}}{2} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k ^{2}} \biggr) \vert u\vert ^{2}\,dx \\& \quad\quad{} +\frac{1}{2(\lambda-\beta_{1})} \int_{\mathbb {R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k ^{2}} \biggr) \bigl\vert g(x, t) \bigr\vert ^{2}\,dx. \end{aligned}$$

And so

$$\begin{aligned}& \frac{d}{dt} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \vert u\vert ^{2}\,dx-2 \nu \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)u \Delta u\,dx+( \lambda-\beta_{1}) \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k ^{2}} \biggr) \vert u\vert ^{2}\,dx \\& \quad\leq2\beta_{3} \int_{\mathbb{R}^{n}}\theta^{2} \biggl( \frac{\vert x\vert ^{2}}{k^{2}} \biggr)a(x)\,dx+\frac{1}{(\lambda-\beta_{1})} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \bigl\vert g(x, t) \bigr\vert ^{2}\,dx. \end{aligned}$$

(3.14)

For the second term on the left-hand side of (3.14), we know

$$\begin{aligned} \begin{aligned}[b] &-\nu \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)u \Delta u\,dx \\ &\quad = \nu \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \vert \nabla u\vert ^{2}\,dx+ \nu \int_{\mathbb{R}^{n}}\theta^{\prime} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \theta \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)u\frac{4x}{k^{2}} \cdot\nabla u\,dx \end{aligned} \end{aligned}$$

(3.15)

and

$$\begin{aligned} \begin{aligned}[b] & \biggl\vert \int_{\mathbb{R}^{n}}\theta^{\prime} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \theta \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr)u\frac{4x}{k^{2}}\cdot\nabla u\,dx \biggr\vert \\ &\quad \leq\frac{4\sqrt{2}c}{k} \int_{k\leq \vert x\vert \leq\sqrt{2}k}\vert u\vert \vert \nabla u\vert \,dx\leq \frac{C}{k}\Vert u\Vert \Vert \nabla u\Vert . \end{aligned} \end{aligned}$$

(3.16)

It follows from (3.15) and (3.16) that

$$\begin{aligned}& \frac{d}{dr} \biggl( e^{\sigma r} \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k ^{2}} \biggr) \vert u\vert ^{2}\,dx \biggr) \\& \quad\leq2\beta_{3}e^{\sigma r} \int_{\vert x\vert \geq k}a(x)\,dx +\frac{e^{\sigma r}}{(\lambda-\beta_{1})} \int_{\vert x\vert \geq k} \bigl\vert g(x, t) \bigr\vert ^{2}\,dx+ \frac{C}{k}e ^{\sigma r}\Vert u\Vert \Vert \nabla u\Vert . \end{aligned}$$

Integrating over the interval $[\tau, t]$, we get

$$\begin{aligned}& \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \bigl\vert u(t) \bigr\vert ^{2}\,dx \\& \quad\leq2\beta_{3}e^{-\sigma t} \int_{\tau}^{t}e^{\sigma r} \int_{\vert x\vert \geq k}a(x)\,dx\,dr +\frac{e^{-\sigma t}}{(\lambda-\beta_{1})} \int_{ \tau}^{t}e^{\sigma r} \int_{\vert x\vert \geq k} \bigl\vert g(x, r) \bigr\vert ^{2}\,dx \,dr \\& \quad\quad{} +\frac{C}{k}e^{-\sigma t} \int_{\tau}^{t}e^{\sigma r}\Vert u\Vert \Vert \nabla u\Vert \,dr+e^{-\sigma t}e^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}, \end{aligned}$$

(3.17)

where $\tau\leq\tau_{0}(t, \hat{\mathscr{D}})$. We now treat each term on the right-hand side of (3.17). For the first term,

$$\begin{aligned} 2\beta_{3}e^{-\sigma t} \int_{\tau}^{t}e^{\sigma r} \int_{\vert x\vert \geq k}a(x)\,dx\,dr \leq2\beta_{3}e^{-\sigma t}e^{\sigma t} \int_{\vert x\vert \geq k}a(x)\,dx\leq2 \beta_{3} \int_{\vert x\vert \geq k}a(x)\,dx, \end{aligned}$$

by (1.4), for any $\varepsilon>0$, there exists $k_{1}( \varepsilon, t)$ such that

$$\begin{aligned} 2\beta_{3} \int_{\vert x\vert \geq k}a(x)\,dx< \frac{\varepsilon}{4} \quad \text{for all } k \geq k_{1}. \end{aligned}$$

(3.18)

For the second term, by (1.5), for any $\varepsilon>0$, there exists $k_{2}(\varepsilon, t)$ such that

$$\begin{aligned}& \frac{1}{(\lambda-\beta_{1})} \int_{\tau}^{t}e^{\sigma r} \int_{\vert x\vert \geq k} \bigl\vert g(x, r) \bigr\vert ^{2}\,dx \,dr \\& \quad \leq\frac{1}{(\lambda-\beta_{1})} \int_{-\infty}^{t}e^{\sigma r} \int_{\vert x\vert \geq k} \bigl\vert g(x, r) \bigr\vert ^{2}\,dx \,dr< \frac{\varepsilon}{4} \quad\text{for all } k\geq k_{2}. \end{aligned}$$

(3.19)

For the forth term, since $u_{\tau}\in B_{0}(\tau)$, by (3.6), for any $t\in\mathbb{R}$, we get

$$\begin{aligned} e^{-\sigma t}e^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}\rightarrow0 \quad \text{as }\tau\rightarrow-\infty. \end{aligned}$$

(3.20)

We now handle the third term on the right-hand side of (3.17). By Young’s inequality, we know

$$\begin{aligned} \frac{C}{k}e^{-\sigma t} \int_{\tau}^{t}e^{\sigma r}\Vert u\Vert \Vert \nabla u\Vert \,dr \leq\frac{C}{2k}e^{-\sigma t} \int_{\tau}^{t}e^{\sigma r}\Vert u\Vert ^{2}\,dr+ \frac{C}{2k}e^{-\sigma t} \int_{\tau}^{t}e^{\sigma r}\Vert \nabla u\Vert ^{2}\,dr. \end{aligned}$$

We can find $\delta_{0}>0$ such that

$$\begin{aligned} \int_{\tau}^{t}e^{\sigma r}\Vert u\Vert ^{2}\,dr\leq \int_{\tau}^{t}e^{( \sigma+\delta_{0})r}\Vert u\Vert ^{2}\,dr, \end{aligned}$$

and by (3.4), we have

$$\begin{aligned}& \int_{\tau}^{t}e^{(\sigma+\delta_{0})r} \bigl\Vert u(r) \bigr\Vert ^{2}\,dr \\& \quad\leq \int_{\tau}^{t}e^{(\sigma+\delta_{0})r} \biggl( \frac{2\beta _{3}\Vert a(x)\Vert _{1}}{\sigma}+e^{-\sigma r}e^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}+\frac{e^{-\sigma r}}{\lambda-\beta_{1}} \int_{-\infty}^{r}e ^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds \biggr) \,dr \\& \quad\leq\frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}e^{(\sigma+\delta_{0})t}+e ^{\sigma\tau} \Vert u_{\tau} \Vert ^{2} \int_{\tau}^{t}e^{\delta_{0}r}\,dr+\frac{1}{ \lambda-\beta_{1}} \int_{\tau}^{t} e^{\delta_{0}s}\,ds \int_{-\infty} ^{t}e^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds \\& \quad\leq\frac{2\beta_{3}\Vert a(x)\Vert _{1}}{\sigma}e^{(\sigma+\delta _{0})t}+\frac{1}{ \delta_{0}}e^{\delta_{0}t}e^{\sigma\tau} \Vert u_{\tau} \Vert ^{2}+\frac{1}{ \delta_{0}(\lambda-\beta_{1})} e^{\delta_{0}t} \int_{-\infty}^{t}e ^{\sigma s} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds \\& \quad< \infty. \end{aligned}$$

Analogously, we can obtain

$$\begin{aligned} \int_{\tau}^{t}e^{\sigma r}\Vert \nabla u\Vert ^{2}\,dr< \infty. \end{aligned}$$

Thus, for any $\varepsilon>0$, there exists $k_{3}(\varepsilon, t)$ such that

$$\begin{aligned} \frac{C}{k}e^{-\sigma t} \int_{\tau}^{t}e^{\sigma r}\Vert u\Vert \Vert \nabla u\Vert \,dr< \frac{ \varepsilon}{4} \quad \text{for all } k\geq k_{2}. \end{aligned}$$

(3.21)

It follows from (3.18)-(3.21) that

$$\begin{aligned} \int_{\vert x\vert \geq2k} \bigl\vert U(t, \tau)u_{\tau} \bigr\vert ^{2}\,dx\leq \int_{\mathbb{R}^{n}}\theta^{2} \biggl(\frac{\vert x\vert ^{2}}{k^{2}} \biggr) \bigl\vert u(t) \bigr\vert ^{2}\,dx< \varepsilon. \end{aligned}$$

So, the proof is complete. □

Next, we utilise Definition 2.6 to prove that the process $\{U(t, \tau)\}_{\tau\leq t}$ associated with the initial value problem (1.1) is $\mathscr{D}$-pullback ω-limit compact.

Lemma 3.4

Assume that (1.2)-(1.4) hold and the external force $g\in L^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$ satisfies (1.5). Then the process $\{U(t, \tau)\}_{\tau \leq t}$ associated with the initial value problem (1.1) is $\mathscr{D}$-pullback ω-limit compact in $L^{2}(\mathbb{R}^{n})$.

Proof

Denote $B_{r}=B(0, r)\cap\mathbb{R}^{n}$, we can split $u(t)$ as

$$\begin{aligned} u(t)=\chi(x)u(t)+ \bigl(1-\chi(x) \bigr)u(t), \end{aligned}$$

where $\chi(x)$ is a smooth function satisfying $0\leq\chi(x) \leq1$, $\vert \chi^{\prime}(x)\vert \leq c_{0}$, and it is defined by

$$\begin{aligned} \chi(x)= \textstyle\begin{cases} 0, &x\in B_{r}, \\ 1, &x\notin B_{r+1}. \end{cases}\displaystyle \end{aligned}$$

And so, we have

$$\begin{aligned}& u_{1}(t)= \textstyle\begin{cases} u(t), &x\in B_{r}, \\ 0, &x\notin B_{r+1}, \\ \chi(x)u(t), &\text{others}, \end{cases}\displaystyle \quad\quad u_{2}(t)= \textstyle\begin{cases} 0, &x\in B_{r}, \\ u(t), &x\notin B_{r+1}, \\ (1-\chi(x))u(t), &\text{others}. \end{cases}\displaystyle \end{aligned}$$

For any $\hat{\mathscr{D}}\in\mathscr{D}\subset B(L^{2}(\mathbb{R} ^{n}))$, $\{U(t,\tau)D(\tau)\}=\{U(t, \tau)u_{\tau}\mid u_{\tau} \in D(\tau)\}$ can be split as

$$\begin{aligned} U(t,\tau)D(\tau)=\chi(x)U(t,\tau)D(\tau)+ \bigl(1-\chi(x) \bigr)U(t,\tau)D( \tau). \end{aligned}$$

By Lemma 2.1, we have

$$\begin{aligned} \alpha \bigl(U(t,\tau)D(\tau) \bigr)\leq\alpha \bigl(\chi(x)U(t, \tau)D(\tau) \bigr)+ \alpha \bigl( \bigl(1-\chi(x) \bigr)U(t,\tau)D(\tau) \bigr). \end{aligned}$$

(3.22)

By Lemma 3.1, we get $u_{1}(t)\in L^{2}(B_{r})$ as $\tau\leq\tau _{0}(t, \hat{\mathscr{D}})$ and

$$\begin{aligned} \chi(x)U(t,\tau)D(\tau)= \bigl\{ \chi(x)U(t,\tau)u_{\tau}=u_{1}(t) \mid u _{\tau}\in D(\tau) \bigr\} . \end{aligned}$$

By Lemma 3.2, we have

$$\begin{aligned} \bigl\Vert u_{1}(t) \bigr\Vert _{H_{0}^{1}(B_{r+1})}= \bigl\Vert \nabla u_{1}(t) \bigr\Vert _{L^{2}(B_{r+1})} \leq R_{1}(t) \quad \text{for all } \tau\leq\tau_{1}(t, \hat{\mathscr{D}}). \end{aligned}$$

Since $H_{0}^{1}(B_{r+1})\hookrightarrow L^{2}(B_{r+1})$ is compact, $\chi(x)U(t,\tau)D(\tau)$ is compact in $L^{2}(B_{r+1})$. By Lemma 2.1, we obtain

$$\begin{aligned} \alpha \bigl(\chi(x)U(t,\tau)D(\tau) \bigr)=0. \end{aligned}$$

(3.23)

By Lemma 3.3, for any $\varepsilon>0$, we can choose r large enough such that

$$\begin{aligned} \int_{\vert x\vert \geq r}\vert u\vert ^{2}\leq\varepsilon. \end{aligned}$$

And then

$$\begin{aligned} \Vert u_{2}\Vert \leq\varepsilon \quad \text{for all }\tau\leq\tau^{\prime}(t, \varepsilon). \end{aligned}$$

(3.24)

We know

$$\begin{aligned} \bigl(1-\chi(x) \bigr)U(t,\tau)D(\tau)= \bigl\{ \bigl(1-\chi(x) \bigr)U(t, \tau)u_{\tau}=u _{2}(t)\mid u_{\tau}\in D(\tau) \bigr\} . \end{aligned}$$

By (3.24), we obtain

$$\begin{aligned} \alpha \bigl( \bigl(1-\chi(x) \bigr)U(t,\tau)D(\tau) \bigr)\leq \varepsilon \quad \text{for all }\tau\leq\tau^{\prime}(t, \varepsilon). \end{aligned}$$

(3.25)

It follows from (3.22), (3.23) and (3.25) that

$$\begin{aligned} \alpha \bigl(U(t,\tau)D(\tau) \bigr)\leq\varepsilon \quad \text{for all } \tau \leq \min \bigl\{ \tau_{0}(t, \hat{\mathscr{D}}), \tau _{1}(t, \hat{ \mathscr{D}}), \tau^{\prime}(t, \varepsilon) \bigr\} . \end{aligned}$$

By Definition 2.6, we obtain $\{U(t, \tau)\}_{\tau\leq t}$ is $\mathscr{D}$-pullback ω-limit compact in $L^{2}(\mathbb{R} ^{n})$. □

Using Theorem 2.2 or Theorem 2.4, it is easy to prove the following theorem by Lemma 3.1 and Lemma 3.4.

Theorem 3.1

Assume that (1.2)-(1.4) hold and the external force $g\in L^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$ satisfies (1.5). Then the process $\{U(t, \tau)\}_{\tau \leq t}$ associated with the initial value problem (1.1) has a $\mathscr{D}$-pullback attractor $\mathcal{A}=\{A(t):t \in\mathbb{R}\}$ in $L^{2}(\mathbb{R}^{n})$.

3.2 $(L^{2}(\mathbb{R}^{n}),L^{p}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback attractors

In this subsection, we prove the existence of $\mathscr{D}$-pullback attractors in $L^{p}(\mathbb{R}^{n})$. We set $\hat{\mathscr{B}}_{1}=\{B_{1}(t): t\in\mathbb{R}\}$, where

$$\begin{aligned} B_{1}(t)= \bigl\{ u\in L^{2} \bigl( \mathbb{R}^{n} \bigr)\cap L^{p} \bigl(\mathbb{R}^{n} \bigr): \Vert u \Vert +\Vert u\Vert _{p}\leq R_{1}(t) \bigr\} \quad \text{for all } t\in\mathbb{R}, \end{aligned}$$

(3.26)

and $R_{1}(t)$ is defined in Lemma 3.2. So by Lemma 3.2, we obtain the family $\hat{\mathscr{B}}_{1}=\{B_{1}(t): t\in\mathbb{R}\}$ is $(L ^{2}(\mathbb{R}^{n}), L^{p}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback absorbing for the process $\{U(t, \tau)\}_{ \tau\leq t}$, i.e. for any $\hat{\mathscr{D}}\in\mathscr{D}\subset B(L^{2}(\mathbb{R}^{n}))$, there exists $\tau_{1}(t, \hat{\mathscr{D}})\leq t$ such that $U(t, \tau)D(\tau)\subset B_{1}(t)$ for all $\tau\leq\tau_{1}(t, \hat{\mathscr{D}})$. We also know

$$\begin{aligned} e^{\sigma t} \bigl( R_{1}(t) \bigr) ^{2} \rightarrow0 \quad \text{as } t\rightarrow-\infty. \end{aligned}$$

(3.27)

Based on Theorem 2.4, we only prove that the process $\{U(t, \tau)\} _{\tau\leq t}$ associated with the initial value problem (1.1) is $\mathscr{D}$-pullback ω-limit compact in $L^{p}(\mathbb{R}^{n})$. Firstly, we prove the following lemma.

Lemma 3.5

Assume that (1.2)-(1.4) hold and the external force $g\in L^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$ satisfies (1.5). Let the family $\hat{\mathscr{B}}_{1}=\{B _{1}(t): t\in\mathbb{R}\}$ be defined by (3.26). Then, for any $\varepsilon\geq0$, any $\hat{\mathscr{D}}\in\mathscr{D}\subset B(L ^{2}(\mathbb{R}^{n}))$ and any $t\in\mathbb{R}$, there exist $M=M(t, \varepsilon)>0$ and $\tau^{\prime\prime}(t, \varepsilon)$ such that

$$\begin{aligned} \begin{aligned}[b] & \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau} \vert \geq M)} \bigl\vert U(t, \tau)u _{\tau} \bigr\vert ^{p}\,dx\leq\varepsilon \\ &\quad \textit{for all } u_{\tau}\in D( \tau), \tau\leq\tau^{\prime\prime }(t, \varepsilon)\textit{ and } M \geq2M_{0}. \end{aligned} \end{aligned}$$

(3.28)

Proof

For any $\varepsilon>0$ be given, by (1.5), there exists $\delta_{1}>0$ such that

$$\begin{aligned} \int_{-\infty}^{t}e^{\sigma s} \int_{e_{1}} \bigl\vert g(x, s) \bigr\vert ^{2}\,dx \,ds< \varepsilon , \end{aligned}$$

(3.29)

where $e_{1}\subset\mathbb{R}^{n}$ and $m(e_{1})\leq\delta_{1}$. By Lemma 2.2 and Lemma 3.2, we know that there exist $M_{1}=M_{1}(t, \varepsilon)$ and $\tau_{2}=\tau_{2}(t, \varepsilon)$ such that

$$\begin{aligned} m \bigl( \mathbb{R}^{n} \bigl( \bigl\vert U(t, \tau)u_{\tau} \bigr\vert \geq M_{1} \bigr) \bigr) \leq \delta_{1}\quad \text{for all } u_{\tau}\in D(\tau) \text{ and } \tau \leq\tau_{2}. \end{aligned}$$

(3.30)

By (1.2) and (1.3), we can choose $M_{2}$ large enough such that

$$\begin{aligned}& \alpha_{1}\vert u\vert ^{p-1}-\beta_{1}\vert u\vert \leq f_{1}(u)\leq\alpha_{2}\vert u\vert ^{p-1}+ \beta_{2}\vert u\vert \quad \text{in } \mathbb{R}^{n}\ \bigl(U(t, \tau)u_{\tau}\geq M_{2} \bigr), \end{aligned}$$

(3.31)

$$\begin{aligned}& \alpha_{3}\vert u\vert ^{p-1}\leq f_{2}(u)\leq \alpha_{4}\vert u\vert ^{p-1} \quad \text{in } \mathbb{R}^{n}\ \bigl(U(t, \tau)u_{\tau}\geq M_{2} \bigr). \end{aligned}$$

(3.32)

Let $M_{0}=\max\{M_{1}, M_{2}\}$ and $\tau\leq\tau_{2}$. Multiplying Eq. (1.1) by $(u-M_{0})_{+}^{p-1}$ and integrating on $\mathbb{R}^{n}$, we have

$$\begin{aligned}& \frac{1}{p}\frac{d}{dt} \int_{\mathbb{R}^{n}} \bigl\vert (u-M_{0})_{+} \bigr\vert ^{p}\,dx- \nu \int_{\mathbb{R}^{n}}\Delta u(u-M_{0})_{+}^{p-1} \,dx+\lambda \int_{\mathbb{R}^{n}}u(u-M_{0})_{+}^{p-1} \,dx \\& \quad{} + \int_{\mathbb{R}^{n}}f_{1}(u) (u-M_{0})_{+}^{p-1} \,dx+ \int_{\mathbb{R}^{n}}a(x)f_{2}(u) (u-M_{0})_{+}^{p-1} \,dx= \int_{\mathbb{R}^{n}}g(x, t) (u-M_{0})_{+}^{p-1} \,dx, \end{aligned}$$

where $(u-M_{0})_{+}$ denotes the positive part of $u-M_{0}$, that is

$$\begin{aligned} (u-M_{0})_{+}= \textstyle\begin{cases} u-M_{0}, &u\geq M_{0}, \\ 0, &u< M_{0}. \end{cases}\displaystyle \end{aligned}$$

Let $\Omega_{1}=\mathbb{R}^{n}(U(t, \tau)u_{\tau}\geq M_{0})$, we get

$$\begin{aligned} (u-M_{0})_{+}^{2p-2}\leq \vert u\vert ^{p-1}(u-M_{0})_{+}^{p-1} \quad \text{and} \quad (u-M_{0})_{+}^{p}\leq u(u-M_{0})_{+}^{p-1} \quad \text{in } \Omega_{1}. \end{aligned}$$

It follows from (3.31), (3.32), Young’s inequality and Hölder’s inequality that

$$\begin{aligned}& -\nu \int_{\mathbb{R}^{n}}\Delta u(u-M_{0})_{+}^{p-1} \,dx=\nu(p-1) \int_{\Omega_{1}}\vert \nabla u\vert ^{2}(u-M_{0})_{+}^{p-2} \,dx\geq0, \end{aligned}$$

(3.33)

$$\begin{aligned}& \begin{aligned}[b] \int_{\mathbb{R}^{n}}f_{1}(u) (u-M_{0})_{+}^{p-1} \,dx&\geq \int_{\Omega _{1}}\alpha_{1}\vert u\vert ^{p-1}(u-M_{0})_{+}^{p-1}\,dx \\ &\quad{} - \int_{\Omega_{1}}\beta _{1}\vert u\vert (u-M_{0})_{+}^{p-1}\,dx, \end{aligned} \end{aligned}$$

(3.34)

$$\begin{aligned}& \int_{\mathbb{R}^{n}}a(x)f_{2}(u) (u-M_{0})_{+}^{p-1} \,dx\geq\alpha_{3} \int_{\Omega_{1}}a(x)\vert u\vert ^{p-1}(u-M_{0})_{+}^{p-1} \,dx\geq0, \end{aligned}$$

(3.35)

$$\begin{aligned}& \begin{aligned}[b] \int_{\mathbb{R}^{n}}g(x, t) (u-M_{0})_{+}^{p-1} \,dx &\leq\frac{1}{2 \alpha_{1}} \int_{\Omega_{1}} \bigl\vert g(x, t) \bigr\vert ^{2}\,dx+ \frac{\alpha_{1}}{2} \int_{\Omega_{1}}(u-M_{0})_{+}^{2p-2}\,dx \\ &\leq\frac{1}{2\alpha_{1}} \int_{\Omega_{1}} \bigl\vert g(x, t) \bigr\vert ^{2}\,dx+ \frac{ \alpha_{1}}{2} \int_{\Omega_{1}}\vert u\vert ^{p-1}(u-M_{0})_{+}^{p-1} \,dx . \end{aligned} \end{aligned}$$

(3.36)

By (3.33)-(3.36), we get

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \int_{\Omega_{1}} \bigl\vert (u-M_{0})_{+} \bigr\vert ^{p}\,dx+(\lambda -\beta_{1}) \int_{\Omega_{1}} \bigl\vert (u-M_{0})_{+} \bigr\vert ^{p}\,dx\leq\frac{1}{2\alpha _{1}} \int_{\Omega_{1}} \bigl\vert g(x, t) \bigr\vert ^{2}\,dx, \end{aligned}$$

which implies that

$$\begin{aligned}& \frac{d}{dt}(t-\tau)e^{\sigma t} \int_{\Omega_{1}} \bigl\vert (u-M_{0})_{+} \bigr\vert ^{p}\,dx+c _{0}e^{\sigma t} \int_{\Omega_{1}} \bigl\vert (u-M_{0})_{+} \bigr\vert ^{p}\,dx \\& \quad \leq\frac{p(t-\tau)}{2\alpha_{1}}e^{\sigma t} \int_{\Omega_{1}}\bigl\vert g(x, t)\bigr\vert ^{2}\,dx, \end{aligned}$$

(3.37)

where $u>0$ in $\Omega_{1}$ and $c_{0}= ( p(\lambda-\beta_{1})- \sigma ) (t-\tau)-1$. Since $\sigma\in(0, \lambda-\beta_{1})$ and $p>2$, there exists $\tau_{3}=\tau_{3}(t, \varepsilon)<0$ such that

$$\begin{aligned} \bigl( p(\lambda-\beta_{1})-\sigma \bigr) (t-\tau)\geq1 \quad \text{for all } \tau\leq\tau_{3}. \end{aligned}$$

So integrating (3.37) over the interval $[\tau, t]$, we have

$$\begin{aligned} \int_{\Omega_{1}} \bigl\vert (u-M_{0})_{+} \bigr\vert ^{p}\,dx\leq\frac{p}{2\alpha_{1}}e^{- \sigma t} \int_{-\infty}^{t}e^{\sigma s} \int_{\Omega_{1}}\bigl\vert g(x, s)\bigr\vert ^{2}\,dx \,ds. \end{aligned}$$

By (3.29), we can obtain

$$\begin{aligned} \int_{\Omega_{1}} \bigl\vert (u-M_{0})_{+} \bigr\vert ^{p}\,dx\leq C\varepsilon \quad\text{for all } \tau\leq \tau_{3} \text{ and } u_{\tau}\in D(\tau), \end{aligned}$$

(3.38)

where $C>0$ is a constant independent of $M_{0}$. Set $\Omega_{2}= \mathbb{R}^{n}(U(t, \tau)u_{\tau}\leq-M_{0})$. Likewise, replacing $(u-M_{0})_{+}$ with $(u+M_{0})_{-}$, we can also obtain that there exists $\tau_{4}=\tau_{4}(t, \varepsilon)$ such that

$$\begin{aligned} \int_{\Omega_{2}} \bigl\vert (u+M_{0})_{-} \bigr\vert ^{p}\,dx\leq C\varepsilon \quad\text{for all } \tau \leq \tau_{4} \text{ and } u_{\tau}\in D(\tau), \end{aligned}$$

(3.39)

where $(u+M_{0})_{-}$ is the negative part of $u+M_{0}$, that is

$$\begin{aligned} (u+M_{0})_{-}= \textstyle\begin{cases} u+M_{0}, &u\leq-M_{0}, \\ 0, &u> -M_{0}. \end{cases}\displaystyle \end{aligned}$$

Then it follows from (3.38) and (3.39) that

$$\begin{aligned} \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau} \vert \geq M_{0})} \bigl\vert \bigl(\vert u\vert -M_{0} \bigr)\bigr\vert ^{p}\,dx \leq\varepsilon \quad\text{for all } \tau \leq \tau^{\prime\prime}(t, \varepsilon) \text{ and } u_{\tau}\in D(\tau), \end{aligned}$$

where $\tau^{\prime\prime}(t, \varepsilon)=\min\{\tau_{3}, \tau _{4}\}$. Hence, we get

$$\begin{aligned}& \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau} \vert \geq2M_{0})} \bigl\vert U(t, \tau)u_{\tau}\bigr\vert ^{p} \,dx \\& \quad= \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau} \vert \geq2M_{0})} \bigl(\vert u\vert -M_{0}+M _{0} \bigr)^{p}\,dx \\& \quad\leq2^{p-1} \biggl( \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau }\vert \geq2M _{0})} \bigl(\vert u\vert -M_{0} \bigr)^{p}\,dx+ \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau} \vert \geq2M_{0})}(M_{0})^{p}\,dx \biggr) \\& \quad\leq2^{p-1} \biggl( \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau }\vert \geq M_{0})}\bigl(\vert u\vert -M _{0} \bigr)^{p}\,dx+ \int_{\mathbb{R}^{n}(\vert U(t, \tau)u_{\tau} \vert \geq M_{0})} \bigl(\vert u\vert -M _{0} \bigr)^{p}\,dx \biggr) \leq2^{p}\varepsilon. \end{aligned}$$

Finally, we obtain (3.28) and the proof is complete. □

By Theorem 2.3, Lemma 3.4 and Lemma 3.5, we can obtain that the process $\{U(t, \tau)\}_{\tau\leq t}$ associated with the initial value problem (1.1) is $\mathscr{D}$-pullback ω-limit compact in $L^{p}(\mathbb{R}^{n})$. So it is easy to prove the following theorem.

Theorem 3.2

Assume that (1.2)-(1.4) hold and the external force $g\in L^{2}_{\mathrm {loc}}(\mathbb{R}, L^{2}(\mathbb{R}^{n}))$ satisfies (1.5). Then the family of sets $\mathcal{A}^{\prime }=\{A^{\prime}(t):t\in\mathbb{R}\}$ is $(L^{2}(\mathbb{R}^{n}), L^{p}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback attractors for $\{U(t, \tau)\}_{\tau\leq t}$.

Proof

We know that the family $\hat{\mathscr{B}}_{1}=\{B_{1}(t): t\in \mathbb{R}\}$ is $(L^{2}(\mathbb{R}^{n}), L^{p}(\mathbb{R}^{n}))$-$\mathscr{D}$-pullback absorbing for the process $\{U(t, \tau)\}_{ \tau\leq t}$, where $B_{1}(t)$ is defined by (3.26). Thus, by Theorem 2.4, we can deduce that the theorem is true. □

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Authors’ Affiliations

(1)

School of Mathematics and Statistics, Southwest University, Chongqing, P.R. China

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Boundary Value Problems

Pullback attractors for a class of non-autonomous reaction-diffusion equations in \(\mathbb{R}^{n}\)

Abstract

Keywords

MSC

1 Introduction

2 Preliminaries

Definition 2.1

Definition 2.2

Definition 2.3

Definition 2.4

Definition 2.5

Lemma 2.1

Definition 2.6

Theorem 2.1

Theorem 2.2

Remark

Theorem 2.3

Theorem 2.4

Remark

Lemma 2.2

Lemma 2.3

3 Main results

3.1 \((L^{2}(\mathbb{R}^{n}),L^{2}(\mathbb{R}^{n}))\)-\(\mathscr{D}\)-pullback attractors

Lemma 3.1

Proof

Lemma 3.2

Proof

Lemma 3.3

Proof

Lemma 3.4

Proof

Theorem 3.1

3.2 \((L^{2}(\mathbb{R}^{n}),L^{p}(\mathbb{R}^{n}))\)-\(\mathscr{D}\)-pullback attractors

Lemma 3.5

Proof

Theorem 3.2

Proof

Declarations

Availability of data and materials

Funding

Authors’ contributions

Ethics approval and consent to participate

Competing interests

Consent for publication

Authors’ Affiliations

References

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