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Attractors for the nonclassical reaction–diffusion equations on time-dependent spaces
Boundary Value Problems volume 2020, Article number: 95 (2020)
Abstract
In this paper, based on the notation of time-dependent attractors introduced by Conti, Pata and Temam in (J. Differ. Equ. 255:1254–1277, 2013), we prove the existence of time-dependent global attractors in \(\mathcal{H}_{t}\) for a class of nonclassical reaction–diffusion equations with the forcing term \(g(x)\in H^{-1}(\varOmega )\) and the nonlinearity f satisfying the polynomial growth of arbitrary \(p-1\) (\(p\geq 2\)) order, which generalizes the results obtained in (Appl. Anal. 94:1439–1449, 2015) and (Bound. Value Probl. 2016: 10, 2016).
1 Introduction
Let Ω be a bounded domain in \(\mathbb{R}^{n}\)\((n\geq 3)\) with smooth boundary, we consider the long-time behavior of the solutions for the following nonclassical reaction–diffusion equation:
where \(t>\tau \), \(\tau \in \mathbb{R}\) is the initial time, \(g(x)\in H^{-1}(\varOmega )\) is an external force term, \(\varepsilon (t)\in C^{1}(\mathbb{R})\) is a decreasing bounded function satisfying
and there exists \(L>0\) such that
For the nonlinear term \(f\in C(\mathbb{R},\mathbb{R})\), similar to that in [3, 20, 24], we make the following classical assumptions:
and
for some positive constants \(c_{0}, c_{1}, c_{2}\).
Let \(\mathcal{F}(u)=\int _{0}^{u}f(r)\,dr\), then there are constants \(\tilde{c}_{i}>0\)\((i=0,1,2)\) such that
For Eq. (1.1), when \(\varepsilon (t)>0\) is a constant, the existence and long-time behavior of solutions have been extensively studied by several authors; see, e.g., [1, 4, 5, 23, 25–27, 29, 30, 32]. In [4, 5, 29], the authors main considered the existence of solutions for this type of equations. In [1, 23, 25–27, 30], the authors main considered the existence of the global attractors (see [23, 25–27]) and the pullback (or the uniform) attractors (see [1, 23, 30]) in \(H_{0}^{1}(\varOmega )\) (or \(H^{1}(\mathbb{R}^{N})\)). In particular, in [32], we obtained the existence of the pullback attractors in \(C_{H_{0}^{1}(\varOmega )}\) (rather than in \(H_{0}^{1}(\varOmega )\)) for the nonclassical reaction–diffusion equations with delays.
When \(\varepsilon (t)=0\), Eq. (1.1) becomes the classical reaction–diffusion equation. The existence and the long-time behavior of solutions have also been extensively investigated by several authors; see, e.g., [2, 11, 12, 17, 21, 28, 31]. In [2, 11, 12, 28], the authors mainly considered the existence (or the blowup), uniqueness and the long-time decay of the solutions for the semilinear parabolic equation [11, 12], the nonlinear parabolic equation [2] and the coupled parabolic systems [28]. In [17, 21, 31], the authors have proved the existence of the global attractors in \(L^{p}(\varOmega )\), \(H_{0}^{1}(\varOmega )\), \(L^{2p-2}(\varOmega )\), \(H^{2}(\varOmega )\) (see [31]) and the existence of the pullback attractors in \(L^{p}(\varOmega )\) and \(H_{0}^{1}(\varOmega )\) (see [17] and [21], respectively).
When \(\varepsilon (t)\in C^{1}(\mathbb{R})\) satisfies (1.2)–(1.3), the long-time behavior of solutions for Eq. (1.1) has been considered by some researchers; see, e.g., [16, 18]. In [16], the authors have proved the existence of the time-dependent global attractors in \(\mathcal{H}_{t}\) with the nonlinearity f satisfying \(|f''(u)|\leq c(1+|u|)\) (see Theorem 3.4 in [16] for details). Furthermore, in [18], the authors have considered the case of the nonlinearity f satisfying the critical exponent growth and proved the existence of the time-dependent global attractors in \(\mathcal{H}_{t}\) (see Theorem 3.3 in [18] for details).
In this paper, we consider Eq. (1.1) with the nonlinearity f satisfying polynomial growth of arbitrary \(p-1\)\((p\geq 2)\) order, which makes that the Sobolev compact embedding is no longer valid and brings more difficulty for verifying the corresponding asymptotic compactness of the solutions process \(\{U(t,\tau )\}_{t\geq \tau }\). In order to overcome the difficulty mentioned above, we verify the existence of the time-dependent global attractors \(\hat{\mathcal{A}}\) in \(\mathcal{H}_{t}\) for the process \(\{U(t,\tau )\}_{t\geq \tau }\) by applying the contractive function methods as in [6, 13, 14, 19, 22, 27] (see Theorem 3.8).
2 Preliminaries
In this section, we firstly review briefly some notations, basic definitions and results about processes on time-dependent spaces (see [7–9, 19] for details).
2.1 Notations
Let \(\{X_{t}\}_{t\in \mathbb{R}}\) be a family of normed spaces, we introduce the R-ball of \(X_{t}\) as
For any given \(\epsilon >0\), the ϵ-neighborhood of a set \(B\subset X_{t}\) is defined as
We denote the Hausdorff semidistance of two (nonempty) sets \(B,C\subset X_{t}\) by
Moreover, we introduce the time-dependent space \(\mathcal{H}_{t}\) endowed with the norms
where \(\|\cdot \|_{2}\) denotes the usual norm in \(L^{2}(\varOmega )\).
2.2 Some concepts
In this subsection, we give some concepts about the time-dependent global attractors.
Definition 2.1
Let \(\{X_{t}\}_{t\in \mathbb{R}}\) be a family of normed spaces. A process is a two-parameter family of mappings \(U(t,\tau ): X_{\tau }\rightarrow X_{t}\), \(t\geq \tau \), \(\tau \in \mathbb{R}\) with properties
- (i)
\(U(\tau,\tau )=\mathrm{Id}\) is the identity operator on \(X_{\tau }\), \(\tau \in \mathbb{R}\);
- (ii)
\(U(t,s)U(s,\tau )=U(t,\tau )\), \(\forall t\geq s\geq \tau \), \(\tau \in \mathbb{R}\).
Definition 2.2
A family \(\hat{C}=\{C_{t}\}_{t\in \mathbb{R}}\) of bounded sets \(C_{t}\subset X_{t}\) is called uniformly bounded if there exists a constant \(R>0\) such that \(C_{t}\subset \mathbb{B}_{t}(R)\) for all \(t\in \mathbb{R}\).
Definition 2.3
A family \(\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}\) is called pullback absorbing if it is uniformly bounded and for every \(R>0\), there exists a constant \(t_{0}=t_{0}(t,R)\leq t\) such that \(U(t,\tau )\mathbb{B}_{\tau }(R)\subset B_{t}\) for all \(\tau \leq t_{0}\).
The process \(\{U(t,\tau )\}_{t\geq \tau }\) is called dissipative whenever it admits a pullback absorbing family.
Definition 2.4
A time-dependent absorbing set for the process \(\{U(t,\tau )\}_{t\geq \tau }\) is a uniformly bounded family \(\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}\) with the following property: for every \(R\geq 0\) there exists a \(t_{0}=t_{0}(R)\geq 0\) such that
Definition 2.5
The process \(\{U(t,\tau )\}_{t\geq \tau }\) is said to be pullback asymptotically compact if for any \(t\in \mathbb{R}\), any bounded sequence \(\{x_{n}\}_{n=1}^{\infty }\subset X_{\tau _{n}}\) and any sequence \(\{\tau _{n}\}_{n=1}^{\infty }\) with \(\tau _{n}\rightarrow -\infty \) as \(n\rightarrow \infty \), the sequence \(\{U(t,\tau _{n})x_{n}\}_{n=1}^{\infty }\) is precompact in \(\{X_{t}\}_{t\in \mathbb{R}}\).
Definition 2.6
The time-dependent global attractor for the process \(\{U(t,\tau )\}_{t\geq \tau }\) is the smallest family \(\hat{\mathcal{A}}=\{\mathcal{A}_{t}\}_{t\in \mathbb{R}}\) such that
- (i)
\(\mathcal{A}_{t}\) is compact in \(X_{t}\);
- (ii)
\(\hat{\mathcal{A}}\) is invariant, i.e., \(U(t,\tau )\mathcal{A}_{\tau }=\mathcal{A}_{t}, \forall t\geq \tau \);
- (iii)
\(\hat{\mathcal{A}}\) is pullback attracting, i.e., it is uniformly bounded and the limit
$$ \lim_{\tau \rightarrow -\infty }\delta _{t}\bigl(U(t,\tau )C_{\tau }, \mathcal{A}_{t}\bigr)=0 $$holds for every uniformly bounded family \(\hat{C}=\{C_{t}\}_{t\in \mathbb{R}}\) and every fixed \(t\in \mathbb{R}\).
Remark 2.7
The attracting property can be equivalently stated in terms of pullback absorbing: a (uniformly bounded) family \(\mathcal{K}=\{K_{t}\}_{t\in \mathbb{R}}\) is called pullback attracting if for any \(\epsilon >0\) the family \(\{\mathcal{O}_{t}^{\epsilon }(K_{t})\}_{t\in \mathbb{R}}\) is pullback absorbing.
Similarly to Theorem 4.2 in [8], we have the following theorem.
Theorem 2.8
The time-dependent global attractor\(\hat{\mathcal{A}}\)exists and it is unique if and only if the process\(\{U(t,\tau )\}_{t\geq \tau }\)is asymptotically compact, namely, the set
is not empty.
2.3 Some results
In order to obtain the time-dependent global attractors of Eq. (1.1), we need the following definitions and conclusions, which are similar to those in [6, 13, 14, 19, 22, 27].
Definition 2.9
Let \(\{X_{t}\}_{t\in \mathbb{R}}\) be a family of Banach spaces and \(\hat{C}=\{C_{t}\}_{t\in \mathbb{R}}\) be a family of uniformly bounded subset of \(\{X_{t}\}_{t\in \mathbb{R}}\). We call a function \(\psi _{\tau }^{t}(\cdot,\cdot )\), defined on \(\{X_{t}\}_{t\in \mathbb{R}}\times \{X_{t}\}_{t\in \mathbb{R}}\), a contractive function on \(C_{\tau }\times C_{\tau }\) if for fixed \(t\in \mathbb{R}\) and any sequence \(\{x_{n}\}_{n=1}^{\infty }\subset C_{\tau }\), there is a subsequence \(\{x_{n_{k}}\}_{k=1}^{\infty }\subset \{x_{n}\}_{n=1}^{\infty }\) such that
We denote the set of all contractive functions on \(C_{\tau }\times C_{\tau }\) by \(Contr(C_{\tau})\).
Theorem 2.10
Let\(\{U(t,\tau )\}_{t\geq \tau }\)be a process on Banach spaces\(\{X_{t}\}_{t\in \mathbb{R}}\)and have a pullback absorbing set\(\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}\). Moreover, assume that, for any\(\epsilon >0\), there exist\(\tau _{0}=\tau _{0}(\epsilon )< t\)and\(\psi _{\tau _{0}}^{t}(\cdot,\cdot )\in \hat{C}(B_{\tau _{0}})\)such that
for any\(t\in \mathbb{R}\). Then\(\{U(t,\tau )\}_{t\geq \tau }\)is pullback asymptotically compact in\(\{X_{t}\}_{t\in \mathbb{R}}\).
Proof
We need to prove that, for any \(\{x_{n}\}_{n=1}^{\infty }\subset B_{\tau _{n}}\) and any \(\tau _{n}\rightarrow -\infty \) as \(n\rightarrow \infty \),
In the following, we will show that \(\{U(t,\tau _{n})x_{n}\}_{n=1}^{\infty }\) has a convergent subsequence via diagonal methods.
Taking \(\epsilon _{m}>0\) with \(\epsilon _{m}\rightarrow 0\) as \(m\rightarrow \infty \).
Then, for \(\epsilon _{1}>0\), by the assumptions, there exist \(\tau _{0}=\tau _{0}(\epsilon _{1})< t\) and \(\psi ^{t}_{\tau _{0}}(\cdot,\cdot )\in \hat{C}(B_{\tau _{0}})\) such that
for any \(t\in \mathbb{R}\), where \(\psi ^{t}_{\tau _{0}}\) depends on \(\tau _{0}\).
Since \(\tau _{n}\rightarrow -\infty \), without loss of generality, we assume that \(\tau _{n}\leq \tau _{0}\) such that \(U(\tau _{0},\tau _{n})x_{n}\in B_{\tau _{0}}\) for each \(n\in \mathbb{N}\). Set \(y_{n}=U(\tau _{0},\tau _{n})x_{n}\), then from (2.1) we have
By the definition of \(\hat{C}(B_{\tau _{0}})\) and \(\psi ^{t}_{\tau _{0}}\in \hat{C}(B_{\tau _{0}})\), we know that \(\{y_{n}\}_{n=1}^{\infty }\) have a subsequence \(\{y_{n_{k}}^{(1)}\}_{k=1}^{\infty }\) such that
Similarly to [13, 22, 27], we have
which, combining with (2.2) and (2.3), implies that
Therefore, there exists a \(K_{1}\in \mathbb{N}\) such that
By induction, we can obtain that, for each \(m\geq 1\), there exists a subsequence \(\{U(t, \tau _{n_{k}}^{(m+1)})x_{n_{k}}^{(m+1)}\}_{k=1}^{\infty }\) of \(\{U(t,\tau _{n_{k}}^{(m)})x_{n_{k}}^{(m)}\}_{k=1}^{\infty }\) and certain \(K_{m+1}\) such that
Now, we consider the diagonal subsequence \(\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=1}^{\infty }\). Since for each \(m\in \mathbb{N}\), \(\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=m}^{\infty }\) is a subsequence of \(\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=1}^{\infty }\), then
which combining with \(\epsilon _{m}\rightarrow 0\) as \(m\rightarrow \infty \), implies that \(\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=1}^{\infty }\) is a Cauchy sequence in \(\{X_{t}\}_{t\in \mathbb{R}}\). This shows that \(\{U(t,\tau _{n})x_{n}\}_{n=1}^{\infty }\) is precompact in \(\{X_{t}\}_{t\in \mathbb{R}}\). □
Similarly to Theorem 3.3 in [19], we have the following conclusion, which will be used to verify the existence of the time-dependent global attractor.
Theorem 2.11
Let\(\{U(t,\tau )\}_{t\geq \tau }\)be a process on Banach space\(\{X_{t}\}_{t\in \mathbb{R}}\), then\(\{U(t,\tau )\}_{t\geq \tau }\)has a time-dependent global attractor in\(\{X_{t}\}_{t\in \mathbb{R}}\)if the following conditions hold:
- (i)
\(\{U(t,\tau )\}_{t\geq \tau }\)has a pullback absorbing set\(\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}\)in\(\{X_{t}\}_{t\in \mathbb{R}}\);
- (ii)
\(\{U(t,\tau )\}_{t\geq \tau }\)is pullback asymptotically compact in\(\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}\).
3 Time-dependent global attractors
In this section, we will establish the existence of the time-dependent global attractors.
3.1 Existence and uniqueness of solutions
In this subsection, we consider the well-posedness of the solutions for Eq. (1.1) with (1.4)–(1.5). At first, we define the weak solutions as follows.
Definition 3.1
A weak solution of Eq. (1.1) is a function \(u\in C([\tau, T]; \mathcal{H}_{t})\cap L^{2}(\tau, T; H_{0}^{1}( \varOmega ))\cap L^{p}(\tau, T; L^{p}(\varOmega ))\) for all \(T>\tau \), with \(u(\tau )=u_{\tau }\) and such that, for all \(\varphi \in H_{0}^{1}(\varOmega )\), it satisfies
Remark 3.2
We notice that, if \(u(t)\) is a weak solution of Eq. (1.1), then it satisfies the energy equality
The following theorem gives the existence of the weak solutions, which is similar to that in [10] and can be obtained by the Faedo–Galerkin methods.
Theorem 3.3
Letfsatisfy (1.4)–(1.5), \(g\in H^{-1}(\varOmega )\)and\(u_{\tau }\in \mathcal{H}_{\tau }\). Then, for any\(\tau \in \mathbb{R}\)and\(t>\tau \), there exists a weak solution\(u(t)\)to Eq. (1.1), which satisfies\(u\in C([\tau,t]; \mathcal{H}_{t})\cap L^{2}(\tau,t;H_{0}^{1}( \varOmega )) \cap L^{p}(\tau,t; L^{p}(\varOmega ))\), \(u_{t}\in L^{2}(\tau,t;\mathcal{H}_{t})\).
Proof
Let \(\{w_{j}\}_{j\geq 1}\subset H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )\) be a Hilbert basis of \(L^{2}(\varOmega )\) such that \(\operatorname{span} \{w_{j}\}_{j\geq 1}\) is dense in \(H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )\). In order to establish the existence of the weak solutions, we need the approximate system for any \(m\geq n\) seeking \(\tilde{u}^{m}(t, x)=\varSigma _{j=1}^{m}\gamma _{mj}(t)\omega _{j}(x)\) that satisfies
for a.e. \(t>\tau, 1\leq j\leq m\).
We will provide a priori estimates that show that these solutions are well-defined in the interval \([\tau,t]\) for any \(t>\tau \).
Step 1: First a priori estimates. Multiplying each equation in the above system by \(\gamma _{mj}(t)\), respectively, and summing from \(j=1\) to m, we obtain
where we have used the Hölder and Young inequalities.
Furthermore, by (1.5), we know that
Integrating it in \([\tau,t]\), we have
Hence,
So, from (3.1), we can get
for all \(t>\tau \).
Moreover, combining with (1.5) and (3.2), we obtain
where \(q=p/(p-1)\).
Then there exist functions \(\tilde{u}\in L^{\infty }(\tau,t; \mathcal{H}_{t}) \cap L^{2}(\tau,t;H_{0}^{1}( \varOmega ))\cap L^{p}(\tau,t;L^{p}(\varOmega ))\) and \(\tilde{\chi }\in L^{q}(\tau, t; L^{q}(\varOmega ))\) for all \(t>\tau \), and a subsequence such that
Step 2: Uniform estimate for the time derivatives. Multiplying each equation of the approximate system by \(\gamma '_{mj}(t)\) and summing from \(j=1\) to m, we arrive at
By the Hölder and Young inequalities, we have
Integrating it from τ to t, and from (1.6) we can get
for all \(t\geq \tau \) and any \(m\geq n\).
Since \(\tilde{u}_{\tau }^{m}=u_{\tau }\) for all \(m\geq n\) and \(\tilde{u}^{m}_{\tau }\in H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )\), by (3.4), we obtain
and
for all \(t>\tau \). Then there exist functions \(\tilde{u}\in L^{\infty }(\tau, t; H_{0}^{1}(\varOmega ) \cap L^{p}( \varOmega ))\) and \(\tilde{u}_{t}\in L^{2}(\tau, t; \mathcal{H}_{t})\) for all \(t>\tau \), which improve the regularity of ũ obtained in Step 1.
For any fixed \(t>\tau \), since
for all \(t_{1},t_{2}\in [\tau,t]\), from (3.5), (3.6) and (3.7), by the Ascoli–Arzelà Theorem, and taking into account the initial data for all the sequence, we deduce that there is a subsequence such that
for all \(t>\tau \) and a.e. in \(\varOmega \times (\tau,\infty )\).
Since \(f\in C(\mathbb{R}, \mathbb{R})\), we conclude that \(f(\tilde{u}^{m})\rightarrow f(\tilde{u}) a.e\). in \(\varOmega \times (\tau, \infty )\). So, combining with (3.3) and [15] (Lemma 1.3, p. 12) we obtain \(\tilde{\chi }=f(\tilde{u})\).
Thus, together with (3.3) and (3.8), by taking the limit in the equations satisfied by \(\{\tilde{u}^{m}\}\) and, thanks to the fact that \(\operatorname{span} \{\omega _{j}\}_{j\geq 1}\) is dense in \(H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )\), we conclude that ũ is a weak solution of Eq. (1.1).
Step 3: Proof of the general statement by density. For each \(n\in \mathbb{N}\), we define \(u^{n}_{\tau }=\varSigma _{j=1}^{n}(u_{\tau }, \omega _{j})\omega _{j}\). (Due to the fact that \(\{\omega _{j}\}_{j\geq 1}\) is a Hilbert basis of \(L^{2}(\varOmega )\), it is easy to check that \(u^{n}_{\tau }\rightarrow u_{\tau }\) in \(\mathcal{H}_{\tau }\).)
Let also consider a sequence \(\{g^{n}\}_{n=1}^{\infty }\subset L^{2}(\varOmega )\) converging to \(g\in H^{-1}(\varOmega )\).
Denote by \(u^{n}\) the corresponding solution to Eq. (1.1) with g replaced by \(g^{n}\) and initial data \(u_{\tau }^{n}\).
Then, by the energy equality for each \(u^{n}\), we have
Similar to the reasoning process in Step 1, we get
for all \(t>\tau \).
Now, combining with (1.5) and (3.9), we see that \(\{f(u^{n})\}\) is bounded in \(L^{q}(\tau,t;L^{q}(\varOmega ))\) for all \(t>\tau \).
Therefore, there exist functions \(u\in L^{\infty }(\tau,t;\mathcal{H}_{t})\cap L^{2}(\tau,t;H_{0}^{1}( \varOmega )) \cap L^{p}(\tau,t;L^{p}(\varOmega ))\) and \(\chi \in L^{q}(\tau,t;L^{q}(\varOmega ))\) for all \(t>\tau \), and a subsequence such that
for all \(t> \tau \).
Moreover, we may improve some of the above convergence. Taking into account the energy equality for \(u^{n}-u^{m}\), we have
By (3.11), we know that
Thus, we have \(u^{n}\rightarrow u\) a.e. in \(\varOmega \times (\tau, \infty )\).
Therefore, as before, combining with (3.10) and [15] (Lemma 1.3, p. 12) we obtain \(\chi =f(u)\); and from (3.10) we may take the limit in the equations satisfied by \(u^{n}\) and conclude that u is a weak solution of Eq. (1.1). □
For the solutions of Eq. (1.1), the following theorem shows the uniqueness and continuity with respect to initial data.
Theorem 3.4
Letfsatisfy (1.4)–(1.5), \(g\in H^{-1}(\varOmega )\)and\(u_{\tau }\in \mathcal{H}_{\tau }\), then the weak solution of Eq. (1.1) is unique. Moreover, for every two solutions\(u^{1}(t)\)and\(u^{2}(t)\) (with different initial data), the following Lipschitz continuity holds:
where\(\omega (t)=u^{1}(t)-u^{2}(t)\).
Proof
Let \(\omega (t)=u^{1}(t)-u^{2}(t)\), then \(\omega (t)\) satisfies the following equation:
Taking the \(L^{2}\)-inner product between (3.12) and ω, and using (1.4), we have
Then
By the Gronwall lemma, it yields
and the uniqueness holds. □
Thus, we define the solution processes \(\{U(t,\tau )\}_{t\geq \tau }\) in the spaces \(\mathcal{H}_{t}\) as:
Moreover, Theorem 3.4 shows that the process \(\{U(t,\tau )\}_{t\geq \tau }\) is Lipschitz in \(\mathcal{H}_{t}\):
3.2 Time-dependent global attractors
In this subsection, we will verify the existence of the time-dependent global attractors in \(\mathcal{H}_{t}\) for the process \(\{U(t, \tau )\}_{t\geq \tau }\) defined by (3.13).
3.2.1 Time-dependent absorbing sets
In the following, we will obtain the time-dependent global absorbing sets.
Lemma 3.5
Letfsatisfy (1.4)–(1.5), \(g\in H^{-1}(\varOmega )\)and\(u_{\tau }\in \mathbb{B}_{\tau }(R)\subset \mathcal{H}_{\tau }\). Then there exists a\(R_{0}>0\)such that the family\(\hat{B}=\{B_{t}(R_{0})\}_{t\in \mathbb{R}}\)is a time-dependent absorbing set for the process\(\{U(t, \tau )\}_{t\geq \tau }\).
Proof
Multiplying (1.1) by \(u(t)\) and integrating over \(x\in \varOmega \), we arrive at
Thanks to (1.5) and the Hölder inequality, we have
Furthermore, by (1.3), we can get
Setting \(\lambda =\min \{\lambda _{1},1\}\) and \(\beta =\frac{\lambda }{1+L}\), we deduce that
Multiplying (3.14) by \(e^{\beta t}\) and integrating it in \([\tau, t]\), we obtain
Therefore,
provided that \(t-\tau \geq t_{0}\) with \(t_{0}=\frac{1}{\beta }\ln (\|u_{\tau }\|_{2}^{2}+\varepsilon (\tau ) \|\nabla u_{\tau }\|_{2}^{2})\), from which we obtain the existence of the time-dependent absorbing set. □
3.2.2 Time-dependent global attractors
At first, we have the following lemma, which is similar to that in [15].
Lemma 3.6
Letfsatisfy (1.4)–(1.5), \(g\in H^{-1}(\varOmega )\), \(u_{\tau }\in \mathcal{H}_{\tau }\)and\(\{u^{n}(t)\}_{n=1}^{\infty }\)be a sequence of solutions for Eq. (1.1) with initial data\(u^{n}_{\tau }\in \mathcal{H}_{\tau }\)\((n=1,2,\ldots )\), then there exists a subsequence of\(\{u^{n}(t)\}_{n=1}^{\infty }\)that converges strongly in\(L^{2}(\tau,t; L^{2}(\varOmega ))\).
Proof
By (1.5) and Theorem 3.3, we know that there exists a sequence \(\{u^{n}(t)\}_{n=1}^{\infty }\subset L^{2}(\tau,T;H_{0}^{1}(\varOmega ))\), \(\{f(u^{n}(t))\}_{n=1}^{\infty }\subset L^{q}(\tau,T;L^{q}(\varOmega ))\). Then, from Eq. (1.1), we obtain \(\partial _{t}u^{n}-\varepsilon (t)\partial _{t}\Delta u^{n}=\Delta u^{n}-f(u^{n})+g(x) \in L^{2}(\tau,T;H^{-1}(\varOmega ))+L^{q}(\tau,T;L^{q}(\varOmega )) \subset L^{2}(\tau,T;H^{-2}(\varOmega ))\). By the regularization theory for elliptic equations, we know that \(\partial _{t}u^{n}\in L^{2}(\tau,T;L^{2}(\varOmega ))\). As in [15], there exists a subsequence of \(\{u^{n}(t)\}_{n=1}^{\infty }\) (still denoted by \(\{u^{n}(t)\}_{n=1}^{\infty }\)) that converges strongly in \(L^{2}(\tau,T; L^{2}(\varOmega ))\). □
Then we have the following theorem, which will obtain the pullback asymptotic compactness for the process \(\{U(t,\tau )\}_{t\geq \tau }\) defined by (3.13).
Theorem 3.7
Letfsatisfy (1.4)–(1.5), \(g\in H^{-1}(\varOmega )\)and\(u_{\tau }\in \mathbb{B}_{\tau }(R)\subset \mathcal{H}_{\tau }\), then\(\{U(t, \tau )\}_{t\geq \tau }\)is pullback asymptotically compact in\(\mathcal{H}_{t}\).
Proof
Let \(u^{i}(t)\) (\(i=1,2\)) be the solutions corresponding to initial data \(u^{i}_{\tau}\in\mathbb{B}_{\tau}(R)\subset\mathcal{H}_{\tau}\), that is, \(u^{i}(t)\) satisfies the following equation:
with initial data
Denoting \(\omega (t)=u^{1}(t)-u^{2}(t)\), then \(\omega (t)\) satisfies the following equation:
with initial data
Multiplying (3.15) by \(\omega (t)\) and integrating it in Ω, then, by (1.4), we obtain
By the Poincaré inequality, we have
where \(\beta _{1}=2\beta \), β is given by (3.14).
Thanks to the Gronwall lemma, we get
Setting
combining with Definition 2.9 and Lemma 3.6, we know that \(\psi_{\tau}^{t}(\cdot,\cdot)\) is a contractive function. Then, for any \(\epsilon >0\) and any fixed \(t\in \mathbb{R}\), let \(\tau _{0}=t-\frac{1}{\beta _{1}}\ln \frac{\|w_{\tau }\|_{2}^{2} +\varepsilon (\tau )\|\nabla w_{\tau }\|_{2}^{2}}{\epsilon }\), we easily see that \(\{U(t,\tau )\}_{t\geq \tau }\) is pullback asymptotically compact in \(\mathcal{H}_{t}\) by Theorem 2.10. □
Combining with Lemma 3.5 and Theorem 3.7, we have the main result of this paper.
Theorem 3.8
Letfsatisfy (1.4)–(1.5), \(g\in H^{-1}(\varOmega )\)and\(u_{\tau }\in \mathbb{B}_{\tau }(R)\subset \mathcal{H}_{\tau }\), then\(\{U(t,\tau )\}_{t\geq \tau }\)possesses a time-dependent global attractor\(\hat{\mathcal{A}}=\{\mathcal{A}_{t}\}_{t\in \mathbb{R}}\)in\(\mathcal{H}_{t}\); that is, \(\mathcal{A}_{t}\)is compact, \(\hat{\mathcal{A}}\)is nonempty, invariant in\(\mathcal{H}_{t}\)and pullback attracts every bounded subset of\(\mathcal{H}_{t}\)with respect to the\(\mathcal{H}_{t}\)-norm.
Remark 3.9
In Theorem 3.8, we have obtained the time-dependent global attractor \(\hat{\mathcal{A}}=\{\mathcal{A}_{t}\}_{t\in \mathbb{R}}\) in \(\mathcal{H}_{t}\). From (1.2) we know that \(\varepsilon (t)\rightarrow 0\) as \(t\rightarrow +\infty \), then Eq. (1.1) becomes the classical reaction–diffusion equation \(u_{t}-\Delta u+f(u)=g(x)\). An interesting question is about the limitation of \(\mathcal{A}_{t}\) as \(t\rightarrow +\infty \), that is, how to describe \(\lim_{t\rightarrow +\infty }\mathcal{A}_{t}\)? We will consider this problem in our next work.
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This work was supported by the NSFC (Grants No. 11601522), the Fundamental Research Funds for the Central Universities of China (Grants No. 17CX02036A) and the Province Natural Science Foundation of Hunan (Grants No. 2018JJ2416).
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Zhu, K., Xie, Y. & Zhou, F. Attractors for the nonclassical reaction–diffusion equations on time-dependent spaces. Bound Value Probl 2020, 95 (2020). https://doi.org/10.1186/s13661-020-01392-7
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DOI: https://doi.org/10.1186/s13661-020-01392-7