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.
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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:
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:
with \(p>2\) and \(\lambda>\beta_{1}\),
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:
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)
 is invariant, i.e.
$$ U(t, \tau)A(\tau)=A({\tau}) \quad\text{for all } t\geq\tau, $$ -
(3)
 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
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
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
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
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
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
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
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
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
Due to (1.2)-(1.4) and Young’s inequality, we get
By (3.2), we obtain
Integrating over the interval \([\tau, t]\) and noting that \(\sigma \in(0, \lambda-\beta_{1})\), we have
Thus, we get
and this implies (3.1). □
Let \(\hat{\mathscr{B}}_{0}=\{B_{0}(t): t\in\mathbb{R}\}\), where
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
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
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
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
Let \(\tau< t-1\) and \(r\in[\tau, t-1]\), integrating over the interval \([r, r+1]\), we get
By (3.4), we find
Thus, by (3.7) and (3.8), we can obtain
Multiplying (1.1) by \(u_{t}\) and integrating on \(\mathbb{R} ^{n}\), we have
And then
It follows from (3.11) that
By (1.5), (3.7), (3.8), (3.10) and the uniform Gronwall inequality, we obtain
It follows from (3.7) and (3.8) that
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
Proof
Choose a smooth function θ such that \(0\leq\theta(s)\leq1\) for \(s\in\mathbb{R}^{+}\),
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
And so
For the second term on the left-hand side of (3.14), we know
and
It follows from (3.15) and (3.16) that
Integrating over the interval \([\tau, t]\), we get
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,
by (1.4), for any \(\varepsilon>0\), there exists \(k_{1}( \varepsilon, t)\) such that
For the second term, by (1.5), for any \(\varepsilon>0\), there exists \(k_{2}(\varepsilon, t)\) such that
For the forth term, since \(u_{\tau}\in B_{0}(\tau)\), by (3.6), for any \(t\in\mathbb{R}\), we get
We now handle the third term on the right-hand side of (3.17). By Young’s inequality, we know
We can find \(\delta_{0}>0\) such that
and by (3.4), we have
Analogously, we can obtain
Thus, for any \(\varepsilon>0\), there exists \(k_{3}(\varepsilon, t)\) such that
It follows from (3.18)-(3.21) that
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
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
And so, we have
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
By Lemma 2.1, we have
By Lemma 3.1, we get \(u_{1}(t)\in L^{2}(B_{r})\) as \(\tau\leq\tau _{0}(t, \hat{\mathscr{D}})\) and
By Lemma 3.2, we have
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
By Lemma 3.3, for any \(\varepsilon>0\), we can choose r large enough such that
And then
We know
By (3.24), we obtain
It follows from (3.22), (3.23) and (3.25) that
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
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
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
Proof
For any \(\varepsilon>0\) be given, by (1.5), there exists \(\delta_{1}>0\) such that
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
By (1.2) and (1.3), we can choose \(M_{2}\) large enough such that
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
where \((u-M_{0})_{+}\) denotes the positive part of \(u-M_{0}\), that is
Let \(\Omega_{1}=\mathbb{R}^{n}(U(t, \tau)u_{\tau}\geq M_{0})\), we get
It follows from (3.31), (3.32), Young’s inequality and Hölder’s inequality that
which implies that
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
So integrating (3.37) over the interval \([\tau, t]\), we have
By (3.29), we can obtain
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
where \((u+M_{0})_{-}\) is the negative part of \(u+M_{0}\), that is
Then it follows from (3.38) and (3.39) that
where \(\tau^{\prime\prime}(t, \varepsilon)=\min\{\tau_{3}, \tau _{4}\}\). Hence, we get
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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Zhang, Q. Pullback attractors for a class of non-autonomous reaction-diffusion equations in \(\mathbb{R}^{n}\) . Bound Value Probl 2017, 146 (2017). https://doi.org/10.1186/s13661-017-0879-5
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DOI: https://doi.org/10.1186/s13661-017-0879-5