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

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.

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. □