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Pullback attractors for non-autonomous reaction-diffusion equation in non-cylindrical domains
Advances in Difference Equations volume 2016, Article number: 230 (2016)
Abstract
In non-cylindrical domains, we use the definition of the variational solution and the maximum principle (the Galerkin method used usually in a cylindrical domain cannot be applied directly) to prove the existence of a pullback \(\mathscr {D}_{\lambda_{1}}\) attractor in \(L^{p}(\mathcal{O}_{t})\) (any \(p\geqslant 2\)) for a reaction-diffusion equation. Then, with an appropriate assumption, the higher-order integrability of variational solution is obtained. Finally, we discuss the existence of a pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{2+\delta}(\mathcal{O}_{t})\) for any \(\delta\in[0,+\infty)\).
1 Introduction
The problems defined in cylindrical domains have been considered extensively. But for some phenomena, such as water waves representing rich phenomena on time-varying spatial domains, for example, the propagation of a tsunami across the open ocean (see [1]), the depth changes as the wave propagates toward the shore, which be seen as an example of a system with time-varying domain. In situations involving rapid temporal changes, such as those occurring during earthquakes or landslides, we may consider a time dependent depth. In other areas, there are a large number of biological processes involving non-cylindrical domains, such as the migrating range of species with season changing, mammalian coat patterns, skin patterns on tropical fish, solid tumor growth etc. Hence, it is necessary to study problems defined in domains varying with time.
Let \(\{\mathcal{O}_{t}\}_{t\in\mathbb{R}}\) be a family of nonempty bounded open subsets of \(\mathbb{R}^{N}\) such that
Define
and
We consider the following initial boundary value problem with homogeneous Dirichlet boundary condition:
where \(u_{\tau}:\mathcal{O}_{\tau}\rightarrow \mathbb{R}\) and \(f:Q_{\tau}\rightarrow\mathbb{R}\) are given for \(\tau\in \mathbb{R}\), and \(g\in C^{1}(\mathbb{R},\mathbb{R})\) satisfies the conditions that there exist nonnegative constants \(\alpha_{1}\), \(\alpha_{2}\), β, l, and \(p\geqslant2\), such that
and
For each \(T>\tau\), consider the auxiliary problem
where \(u_{\tau}:\mathcal{O}_{\tau}\rightarrow \mathbb{R}\) for \(\tau\in\mathbb{R}\), g satisfies (1.4)-(1.5), and \(f:Q_{\tau,T}\rightarrow\mathbb{R}\).
When the domains vary with respect to time, even though the forcing term f is independent of time, the problem is still non-autonomous.
For domains \(\mathcal{O}_{t}\equiv\mathcal{O}\), \(t\in\mathbb{R}\), (1.3) returns to the usual non-autonomous reaction-diffusion equation which was studied extensively (see, for example, [2–7]). Specially, we have [5] obtained a pullback attractor in \(L^{p}(\mathcal{O})\) (p comes from (1.4)) by requiring
where \(0\leqslant\alpha<\lambda\), λ is the first eigenvalue of −Δ in \(H_{0}^{1}(\mathcal{O})\). Recently, Łukaszewicz in [4] applying the Galerkin method and the Gronwall lemma introduced in [3] with forcing terms \(f\in L^{2}_{\mathrm{loc}}(\mathbb{R}; L^{2}(\mathcal{O}))\) and
considered the existence of pullback \(\mathscr{D}\) attractor in \(L^{p}\), which, to the best of our knowledge, is the best result.
For the case of domains changing in time, one may refer to [8–14] and the references therein. Specially, for domains expanding in time (that is, the condition (1.1)), by using a penalty method, the authors in [10] considered the existence and uniqueness of a variational solution for (1.3) and obtained \((L^{2}(\mathcal {O}_{\tau}), L^{2}(\mathcal {O}_{t}))\) pullback \(\mathscr{D}_{\lambda_{1}}\) attractor. By applying a diffeomorphism, [11] obtained the existence and uniqueness of a weak solution for (1.3), furthermore, one proved the existence of a \((L^{2}(\mathcal {O}_{\tau}), L^{2}(\mathcal {O}_{t}))\) pullback \(\mathscr{D}\) attractor.
Then, as the problem defined in a cylindrical domain, we may ask whether the pullback attractor in \(L^{p}(\mathcal {O}_{t})\) exists as well; moreover, for any \(\delta\geqslant0\), what is the result as regards the pullback attractor in \(L^{2+\delta}(\mathcal {O}_{t})\)?
In cylindrical domains, by applying the Garlerkin method and selecting \(\vert (u-M)_{+}\vert ^{p-2}(u-M)_{+}\) as a testing function, the pullback attractor in \(L^{p}\) has been proved (see [4–6]). But in non-cylindrical domains, due to the characteristic of the penalty term (which is defined in Hilbert space), it is difficult or even impossible to choose \(\vert (u-M)_{+}\vert ^{p-2}(u-M)_{+}\) as a testing function, so the method in [4] cannot be applied directly to deduce the existence of pullback attractor in \(L^{p}(\mathcal {O}_{t})\). On the other hand, when we have a forcing term \(f\in L^{2}_{\mathrm{loc}}(\mathbb{R};L^{2}(\mathcal {O}_{t}))\), it is impossible to obtain a solution \(u\in L^{2+\delta}(\mathcal {O}_{t})\) for any \(\delta\leqslant0\). Based on the existence of the \(L^{2}(\mathcal {O}_{t})\) pullback \(\mathscr{D}_{\lambda_{1}}\) attractor, the authors in [13] provided a new method to show that the pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{2}(\mathcal {O}_{t})\) can attract sets in the topology of \(L^{2+\delta}(\mathcal {O}_{t})\) (\(\forall\delta\in [0,\infty)\)). But the existence of a pullback \(\mathscr{D}_{\lambda _{1}}\) attractor in \(L^{2+\delta}(\mathcal {O}_{t})\) is still unknown.
In this paper, by applying the definition of a variational solution and a maximum principle, we first consider the existence of a pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{p}(\mathcal {O}_{t})\); then, as a corollary, with the forcing term satisfying \(f\in L^{\infty }_{\mathrm{loc}}(\mathbb{R}^{N+1})\), we establish the existence of the pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{2+\delta}(\mathcal {O}_{t})\) (\(\delta \in[0, +\infty)\)) for a non-autonomous reaction-diffusion equation defined in non-cylindrical domains. The main results are the following.
Theorem 1.1
Suppose (1.1), (1.2), (1.4), and (1.5) hold, \(u_{\tau}\in L^{2}(\mathcal {O}_{\tau})\) and \(f\in L^{2}_{\mathrm{loc}}(\mathbb{R}^{N+1})\) satisfying
here \(\lambda_{1,t}\) is the first eigenvalue of −Δ in \(H_{0}^{1}(\mathcal {O}_{t})\). Then the process \(U(\cdot,\cdot)\) corresponding to problem (1.3) has a minimal unique pullback \(\mathscr{D}_{\lambda _{1}}\) attractors \(\mathscr{A}(t)\) in \(L^{p}(\mathcal {O}_{t})\) (p comes from (1.4)) for all \(t\in\mathbb{R}\).
Theorem 1.2
Suppose that (1.1), (1.2), (1.4), and (1.5) hold, \(u_{\tau}\in L^{2}(\mathcal {O}_{\tau})\) and \(f\in L^{\infty}_{\mathrm{loc}}(\mathbb {R}^{N+1})\). Let \(U(\cdot,\cdot)\) be the process generated by the variational solutions of (1.3). Then, for any \(\delta\in[0,\infty)\), there exists a pullback \(\mathscr{D}_{\lambda _{1}}\) attractor \(\hat{\mathscr{A}}\) in \(L^{2+\delta}(\mathcal {O}_{t})\), that is, \(\hat{\mathscr{A}}\) is compact, invariant and we have a pullback \(\mathscr{D}_{\lambda_{1}}\) attracting in the topology of \(L^{2+\delta}(\mathcal {O}_{t})\) for any \(\delta\in[0,+\infty)\).
This paper is organized as follows. In Section 2, we will present some definitions, and related results as regards the pullback attractor and a variational solution. By means of the definition of the variational solution and the maximum principle, in Section 3, the pullback \(\mathscr {D}_{\lambda_{1}}\) attractor in \(L^{p}(\mathcal {O}_{t})\) is established (see Theorem 1.1). In Section 4, the higher-order integrability of the approximation solution and the variational solution associated to (1.3) are obtained. Furthermore, we discuss the existence of a pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{2+\delta}(\mathcal {O}_{t})\) (\(\delta\in [0,+\infty)\)) (see Theorem 1.2).
2 Preliminaries
For each \(t\in\mathbb{R}\), we denote by \((\cdot,\cdot)_{t}\) and \(\vert \cdot \vert _{t}\) the usual inner product and related norm in \(L^{2}(\mathcal {O}_{t})\) and by \(((\cdot,\cdot))_{t}\) and \(\|\cdot\|_{t}\) the usual gradient inner product and associated norm in \(H^{1}_{0}(\mathcal {O}_{t})\). The usual duality product between \(H^{1}_{0}(\mathcal {O}_{t})\) and \(H^{-1}(\mathcal {O}_{t})\) will be denoted by \(\langle\cdot,\cdot\rangle_{t}\). Also \((\cdot,\cdot)_{t}\) and \(\|\cdot\|_{L^{p}(\mathcal {O}_{t})}\) represent the duality product between \(L^{p}(\mathcal {O}_{t})\) and \(L^{q}(\mathcal {O}_{t})\) with \(\frac{1}{p}+\frac{1}{q}=1\) and associated norm.
2.1 Pullback attractor
We will recall some concepts about pullback attractor which can be found, for example, in [6, 10].
We consider a process (also called a two-parameter semigroup) U on a Banach space X, i.e. a family \(\{U(t,\tau);-\infty<\tau \leqslant t<+\infty\}\) of continuous mappings \(U(t,\tau):X\to X\), such that
Suppose \(\mathscr{D}\) is a nonempty class of parameterized sets \(\hat{D}=\{D(t):t\in\mathbb{R}\}\subset\mathcal{P}(X)\), where \(\mathcal{P}(X)\) denotes the family of all nonempty subsets of X.
Definition 2.1
It is said that \(\hat{B}\in\mathscr{D}\) is pullback \(\mathscr{D}\)-absorbing for the process \(U(\cdot,\cdot)\) if for any \(t\in\mathbb{R}\) and any \(\hat{D}\in\mathscr{D}\), there exists a \(\tau_{0}=\tau_{0}(t,\hat{D})\leqslant t\) such that
Definition 2.2
The process \(U(\cdot,\cdot)\) is called norm-to-weak continuous if
for all \(t\geqslant\tau\), \(\tau\in\mathbb{R}\), where X is a Banach space.
Let X be a complete metric space and B be a bounded subset of X. The Kuratowskii measure of noncompactness \(\alpha(B)\) of B is defined by
Definition 2.3
A process \(\{U(t,\tau)\}\) is called pullback ω-\(\mathscr {D}\)-limit compact if for any \(\epsilon>0\) and \(\hat{D}\in\mathscr{D}\), there exists a \(\tau_{0}(\hat{D},t)\leqslant t\) such that \(\alpha(\bigcup_{\tau\leqslant\tau_{0}}U(t,\tau)D(\tau))\leqslant \epsilon\).
Theorem 2.4
Let \(U(t,\tau)\) be a process in X satisfying the following conditions:
-
(i)
\(U(t,\tau)\) is norm-to-weak continuous in X;
-
(ii)
there exists a family B̂ of pullback \(\mathscr {D}\)-absorbing sets in X;
-
(iii)
\(U(t,\tau)\) is pullback ω-\(\mathscr{D}\)-limit compact in X.
Then there exists a minimal pullback \(\mathscr{D}\)-attractor \(\mathscr{\hat {A}}\) in X given by
By following a similar process to Lemma 2.1 in [4], we can get the result defined in non-cylindrical domains; it is very useful for checking condition (iii) in Theorem 2.4 when the phase space is a Lebesgue space \(L^{p}(\mathcal {O}_{t})\).
Theorem 2.5
Let \(U (t, \tau) \) be a process in \(L^{p}(\mathcal {O}_{t})\) and \(L^{q}(\mathcal {O}_{t})\) respectively, \(p> q> 0\), satisfying the following conditions:
-
(i)
\(U (t, \tau) \) is pullback ω-\(\mathscr{D}\)-limit compact in \(L^{q}(\mathcal {O}_{t})\);
-
(ii)
for every \(\varepsilon> 0 \) and \(\hat{B}\in\mathscr{D}\), there exist \(M = M (\varepsilon,\hat{B}) \) and \(\tau_{1}=\tau_{1}(\varepsilon,\hat{B})\) such that
$$\int_{\vert U(t,\tau)u_{\tau} \vert \geqslant M}\bigl| U(t,\tau)u_{\tau}\bigr| ^{p}\,dx< \varepsilon $$for any \(u_{\tau}\in B(\tau)\), \(\tau\leqslant\tau_{1}\) and any \(t\in\mathbb{R}\).
Then \(U(t,\tau) \) is pullback ω-\(\mathscr{D}\)-limit compact in \(L^{p}(\mathcal {O}_{t})\).
Theorem 2.6
([15])
Let X, Y be two Banach spaces, \(X^{*}\), \(Y^{*}\) be respectively their dual spaces. Assume that X is dense in Y, the injection \(i : X \rightarrow Y \) is continuous, its adjoint \(i^{*} : Y^{*} \rightarrow X ^{*} \) is dense, and U is a norm-to-weak continuous process on Y. Then U is a norm-to-weak continuous process on X if and only if, for any \(\tau\in\mathbb{R}\), \(t \geqslant\tau\), \(U (t, \tau) \) maps compact sets of X to bounded sets of X.
2.2 Variational solutions for equation (1.6)
As it is convenient for the application later, we recall some notations and variational solutions of (1.6) given by [10].
For each \(T>\tau\), denote
Definition 2.7
A variational solution of equation (1.6) is a function u such that
Theorem 2.8
([10])
Suppose that (1.1), (1.2), (1.4), and (1.5) hold, for \(f\in L^{2}(\tau, T;L^{2}(\mathcal {O}_{T}))\) and \(u_{\tau}\in L^{2}(\mathcal {O}_{\tau})\), there exists a unique variational solution \(u\in L^{2}(\tau, T;H_{0}^{1}(\mathcal {O}_{T}))\cap L^{p}(\tau,T;L^{p}(\mathcal {O}_{T}))\) of equation (1.6) and the variational solution satisfies the energy equality a.e. \(t\in[\tau,T]\), that is,
holds for a.e. \(t\in[\tau,T]\). In addition, \(u\in C([\tau,T]; L^{2}(\mathcal {O}_{T}))\) and satisfies the energy equality for all \(t\in[\tau,T]\). Moreover, if \(u_{\tau}\in H_{0}^{1}(\mathcal {O}_{\tau})\cap L^{p}(\mathcal {O}_{\tau})\), then u also satisfies
Definition 2.9
([10])
\(u:Q_{\tau}\to\mathbb{R}\) is called a variational solution of (1.3), if, for any \(T>\tau\), u restricted in \(Q_{\tau,T}\) is the variational solution of (1.6).
Let
Suppose f satisfies (2.2), from Theorem 2.8 and Definition 2.9, for any \(\tau\in\mathbb{R}\), any \(u_{\tau }\in\L^{2}(\mathcal {O}_{\tau})\) and a.e. \(t\in(\tau,T)\) (\(T>\tau\)), there exists a unique variational solution \(u(\cdot;\tau,u_{\tau})\).
Let
then (2.3) defines a process \(U(\cdot,\cdot)\) of a family of set \(\{u_{\tau}\in L^{2}(\mathcal {O}_{\tau}), \tau\in\mathbb{R}\}\).
Let \(\mathcal{R}_{\lambda_{1}}\) be the set of functions \(r:\mathbb{R}\to [0,\infty)\) satisfying
where \(\lambda_{1,t}\) is the first eigenvalue of −Δ in \(H_{0}^{1}(\mathcal {O}_{t})\). Denote by \(\mathscr{D}_{\lambda_{1}}\) the family of \(\hat{D}:=\{D(t)\subset L^{2}(\mathcal {O}_{t}), t\in\mathbb{R}\}\) such that
here \(\bar{B}(0,r_{\hat{D}}(t))\) is the closed ball with center in 0 and radius \(r_{\hat{D}}(t)\).
Then, the following result holds.
Lemma 2.10
([14])
Suppose (1.1), (1.2), (1.4), and (1.5) hold, \(u_{\tau}\in L^{2}(\mathcal {O}_{\tau})\) and f satisfies (2.2). Then, process \(U(\cdot,\cdot)\) associated to equation (1.3) generates a unique \((L^{2}(\mathcal {O}_{\tau}),L^{2}(\mathcal {O}_{t}))\) pullback \(\mathscr{D}_{\lambda_{1}}\) attractor \(\hat{\mathscr{A}}\).
Together with Theorem 2.5 and Theorem 2.6, we easily obtain the pullback \(\mathscr{D}_{\lambda _{1}}\) attractor in \(L^{p}\) (\(p>2\)) defined in domains increasing with time.
Theorem 2.11
Let \(U(t,\tau)\) be a process in \(L^{p}(\mathcal {O}_{t})\) satisfying the following conditions:
-
(i)
\(U(t,\tau)\) is norm-to-weak continuous in \(L^{p}(\mathcal {O}_{t})\);
-
(ii)
there exists a family B̂ of pullback \(\mathscr{D}_{\lambda _{1}}\)-absorbing sets in \(L^{p}(\mathcal {O}_{t})\);
-
(iii)
\(U(t,\tau)\) is pullback ω-\(\mathscr{D}_{\lambda_{1}}\)-limit compact in \(L^{p}(\mathcal {O}_{t})\).
Then there exists a minimal pullback \(\mathscr{D}_{\lambda_{1}}\)-attractor \(\mathscr{\hat{A}}\) in \(L^{p}(\mathcal {O}_{t})\) given by
To make the test function used later meaningful, in [14], the following maximum principle holds.
Lemma 2.12
([14])
Suppose \(u_{\tau}\in L^{\infty}(\mathcal {O}_{\tau})\cap H_{0}^{1}(\mathcal {O}_{\tau})\), \(f \in L^{\infty}(\tilde{Q}_{\tau, T})\), and g satisfies (1.4). Then there exists a positive constant K depending on \(\Vert u_{\tau} \Vert _{L^{\infty }(\mathcal {O}_{\tau})}\), \(\Vert f\Vert _{L^{\infty}(\tilde{Q}_{\tau,T})}\), β, and \(\alpha_{1}\) such that the variational solution u of equation (1.6) satisfies
3 Pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{p}\)
In this section, by applying the definition of the variational solution of equation (1.6), we consider the existence of a pullback \(\mathscr {D}_{\lambda_{1}}\) attractor in \(L^{p}\) (p satisfying (1.4)).
Denote
Lemma 3.1
Assume \(u_{\tau m} \in L^{\infty}(\mathcal {O}_{\tau})\cap H_{0}^{1}(\mathcal {O}_{\tau})\), \(f_{m} \in L^{\infty}_{\mathrm{loc}}(\mathbb{R}, L^{\infty}(\mathcal {O}_{t}))\), then the variational solution \(u_{m}(t)\) of (1.6) satisfies the inequality
for a.e. \(t\in(\tau,T)\).
Proof
For any fixed \(\tau\in(-\infty, T]\), denoted by \(u_{m}(t)\) we have the variational solution of equation (1.6) corresponding to data \((u_{\tau m}, f_{m})\) with
By Lemma 2.12 and Theorem 2.8, we know
and for any \(\eta\in C^{1}_{c}(\tau,T)\), \(\eta u_{m}\in\mathcal{U}_{\tau,T}\). According to the details of Lemma 2.12 (introduced in [14]), that is, denoting
and by (1.4), we know there exists a positive constant \(K'\) depending on nonnegative constants β, \(\alpha_{1}\), and a positive constant M, such that
Defining
we have \(\Vert u_{m}\Vert _{L^{\infty}(\tilde{Q}_{\tau,T})}\leqslant K\). Hence, we obtain
for the M above and \(p>2\), and for any \(\eta\in C_{c}^{1}(\tau,T)\),
So, according to the definition of a variational solution, we can choose
as a testing function to obtain
therefore, for a.e. \(t\in(\tau, T)\),
Notice
By (1.4), as M is large enough, we know \(g(u_{m}(t))\geqslant\frac {\alpha_{1}}{2}\vert u_{m}(t)\vert ^{p-1}\), hence,
Inserting (3.5) and (3.6) into (3.4), for a.e. \(t\in(\tau, T)\), we deduce
for a.e. \(t\in(\tau,T)\) holds.
Similarly, replacing \((u_{m}(t)-M)_{+}\) with \((u_{m}(t)+M)_{-}\) and following a similar process, we obtain
for a.e. \(t\in(\tau,T)\).
Combining inequalities (3.7) and (3.8), we see that (3.1) holds for a.e. \(t\in(\tau,T)\). □
The following Gronwall lemma, introduced by Łukaszewicz in [3], is useful in an inequality for estimating.
Lemma 3.2
([3])
For some \(\lambda>0\), \(\tau\in\mathbb{R}\) and \(s>\tau\), let
where functions y, \(y'\), h are assumed to be locally integrable and y, h are nonnegative on the interval \(t< s< t+r\), for some \(t\geqslant\tau\). Then
In cylindrical domains, Lemma 3.3 and Lemma 3.4 in [4] have provided the method checking the conditions of asymptotic compact in \(L^{p}(\mathcal {O})\) (\(p>2\)), and following a similar idea, we establish the lemma in non-cylindrical domains.
Lemma 3.3
Let \(\hat{D}=\{D(\tau):\tau\in\mathbb{R}\}\in \mathscr{D}_{\lambda_{1}}\). Then, for any \(\varepsilon>0\) and each \(t\in \mathbb{R}\), there exist constants M and \(\tau_{1}=\tau_{1}(t, \hat{D})\) such that
for any \(\tau\leqslant \tau_{1}\), any \(u_{\tau}\in\hat{D}\). Furthermore, there exists a constant \(\bar{M}>0\) such that
for any \(\tau\leqslant \tau_{1}\) and any \(u_{\tau}\in\hat{D}\).
Proof
Assume that \(u_{\tau m}\in L^{\infty}(\mathcal {O}_{\tau})\cap H_{0}^{1}(\mathcal {O}_{\tau})\) and \(f_{m}\in L^{\infty}(\tilde{Q}_{\tau,T})\) for any fixed \(\tau\leqslant T\), \(T \in\mathbb{R}\). Then for \(u_{\tau}\in L^{2}(\mathcal {O}_{\tau})\) and \(f\in L^{2}_{\mathrm{loc}}(\mathbb{R};L^{2}(\mathcal {O}_{t}))\), there exist sequences \(u_{\tau m}\in L^{\infty}(\mathcal {O}_{\tau})\cap H_{0}^{1}(\mathcal {O}_{\tau})\), \(f_{m}\in L^{\infty}(\tilde{Q}_{\tau,T})\), such that
and, for each \(m=1,2,\ldots\) ,
Applying the Gronwall Lemma 3.2 to (3.1) and combining with (3.10), we obtain for \(r>0\)
Noting \(\eta u_{m}\in\mathcal{U}_{\tau,T}\) (any \(\eta\in C_{c}^{1}(\tau ,T)\)), we have
in addition, for a.e. \(t\in(\tau,T)\),
By using (1.4) and the Cauchy inequality, it follows that
and, in particular, that
Integrating the inequality (3.13) over \([t,t+\frac{r}{2}]\), we have
for any \(t\in(\tau,T)\). In addition, applying the Gronwall lemma to (3.13), we also have
for any \(t\in[\tau,T]\), where \(\lambda_{1,T}\) is the first eigenvalue of −Δ in \(H_{0}^{1}(\mathcal {O}_{T})\).
Combining (3.14), (3.15), and (3.11), we know that
where \(C_{1}\) is a constant independent of m, and depending on \(\frac {1}{\alpha_{1}}\), \(e^{-\lambda_{1,T}(t-\tau)}\vert u_{\tau} \vert ^{2}_{\tau}\), \(\int _{-\infty}^{t} e^{\lambda_{1,T}s}\vert f(s)\vert _{T}^{2}\,ds\), \(\frac{2\beta \vert \mathcal {O}_{T}\vert }{\lambda_{1,T}}\), \(\int_{t}^{t+\frac{r}{2}}\vert f(s)\vert _{T}^{2}\,ds\).
So, \((\vert u_{m}(t+r)\vert -M)_{+}\) is bounded in \(L^{\infty}(\tau, T; L^{p}(\mathcal {O}_{T}))\) and there exists a subsequence denoted still by \((\vert u_{m}(t+r)\vert -M)_{+}\) such that
Hence,
for the \(C_{1}\) above.
For any \(t\in\mathbb{R}\), \(\varepsilon>0\), and any \(\hat{\mathcal{D}} \in\mathscr{D}_{\lambda_{1}}\), there exist constants \(\tau_{1}(t, \hat{\mathcal{D}})\) and \(M_{1}\) such that for \(M\geqslant M_{1}\) and \(\tau <\tau_{1}(t, \hat{\mathcal{D}})\), one obtains
moreover, with M large enough and the function f integrable on \([t,t+r]\), we have
Therefore,
for \(M\geqslant M_{1}\) and \(\tau<\tau_{1}(t, \hat{\mathcal{D}})\).
Finally, let \(\bar{M}:=2M\), and by (3.9) we conclude that
for any \(\tau\leqslant \tau_{1}(t,\hat{\mathcal{D}})\) and any \(u_{\tau}\in\hat{D}\). □
Now, we prove Theorem 1.1.
According to Lemma 2.10, Theorem 2.5, and Lemma 3.3, it is sufficient to check the existence of a pullback \(\mathscr{D}_{\lambda_{1}}\) absorbing set in \(L^{p}\).
By Lemma 3.3, for any \(u_{\tau}\in\hat{D}\) and any \(\tau\leqslant \tau_{1}(t,\varepsilon,\hat{D})\), we have
4 Pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{2+\delta}\)
Throughout this section, the T will be required to obey \(T\geqslant t\) or \(T\geqslant s+1\). We first consider the higher-order integrability for approximation solutions and variational solutions of (1.6), respectively, then we obtain a pullback \(\mathscr{D}_{\lambda_{1}}\) absorbing set in \(L^{2+\delta}(\mathcal {O}_{t})\) (\(\forall\delta\in[0,\infty)\)). Finally, we prove the existence of a pullback \(\mathscr{D}_{\lambda_{1}}\) attractor in \(L^{2+\delta}(\mathcal {O}_{t})\) for any \(\delta\in[0,\infty)\).
4.1 Higher-order integrability for approximation solutions
For any \(\hat{D}=\{D(t): t\in\mathbb{R}\}\in\mathscr{D}_{\lambda_{1}}\) and \(u_{\tau}\in D(\tau)\), set \(u(t)=U(t,\tau)u_{\tau}\). For any fixed \(\tau\in(-\infty, T]\), denoted by \(u_{m}(t)\), we have the variational solution of equation (1.6) corresponding to the data \((u_{\tau m}, f)\) satisfying
By Lemma 2.12 and Theorem 2.8, we know
and
for any \(\phi\in\mathcal{U}_{\tau,T}\).
For any \(\theta\geqslant0\), we still have
and, for any \(\eta\in C_{c}^{1}(\tau,T)\),
Hence, we can choose \(\eta \vert u_{m}\vert ^{\theta}u_{m}\) as a test function to obtain
moreover,
Therefore, for a.e. \(t\in(\tau, T)\),
By (1.4), we know
Noticing (3.3), we can select an appropriate \(K'\) such that
So,
Combining (4.2), (4.3), and (4.5), we deduce
for a.e. \(t\in(\tau,T)\). Due to \(u_{m}(t)\in H_{0}^{1}(\mathcal {O}_{t})\) and
it follows from (4.6) that
We prove the following result.
Theorem 4.1
Let \(\hat{D}\in\mathscr{D}_{{\lambda}_{1}}\) and T be a fixed time. Assume further that the initial data \(u_{\tau m }\) and f satisfy (4.1). Then, for each \(t\in(\tau,T)\) and each \(h=0,1,2,\ldots\) , there exist two sequences (positive constants) \(\tilde{T}_{h}(t,\hat{D})\) (depending only on h, t, \(\lambda_{1,T}\) and D̂) and \(\tilde{M}_{h}(t)\) (depending only on t, h, \(\lambda_{1,T}\), N, \(\vert \mathcal {O}_{T}\vert \) and \(\Vert f(t)\Vert _{L^{\infty}(\mathcal {O}_{T})}\)), such that for any \(m=1,2,\ldots\) , the variational solution \(u_{m}\) of (1.6) satisfies
and
for any \(s-\tau\geqslant \tilde{T}_{h}(t,\hat{D})\).
Proof
Selecting \(\theta=0\) in (4.2), for a.e. \(t\in(\tau,T)\), we have
By using (1.4) and the Cauchy inequality, it follows that
and, in particular, that
moreover,
for any \(t\in[\tau,T]\), where \(\lambda_{1,T}\) is the first eigenvalue of −Δ in \(H_{0}^{1}(\mathcal {O}_{T})\).
For each \(t\in\mathbb{R}\), let
combined with (4.11) to get
Noticing that \(u_{\tau}\in D(\tau)\) with \(\hat{D}=\{D(t): t\in \mathbb{R}\}\in\mathscr{D}_{\lambda_{1}}\), by (4.12), there exists a constant \(T'(t,\hat{D})\) such that
Taking \(\theta=0\) in (4.7) and integrating with respect to t, we obtain
On the other hand, from the embedding \(H_{0}^{1}(\mathcal {O}_{T}) \hookrightarrow L^{\frac{2N}{N-2}}(\mathcal {O}_{T})\), we know that there is a constant \(c_{N}\) such that
From (4.14) and (4.15), it follows that
Setting
and by (4.13), (4.16), we know that (\(A_{0}\)) and (\(B_{0}\)) hold.
In the following, performing induction, we assume (\(A_{h}\)) and (\(B_{h}\)) hold to prove (\(A_{h+1}\)) and (\(B_{h+1}\)).
Taking \(\theta=2(\frac{N}{N-2})^{h+1}-2\) in (4.7), we obtain
moreover,
Applying the uniform Gronwall lemma to (4.18) and by (\(B_{h}\)) we get
For any \(s-\tau\geqslant\tilde{T}_{h}(t,\hat{D})+1\), we integrate (4.17) over \([s,s+1]\) and obtain
Since \(u_{m}\) is a variational solution of (1.6) and bounded in \(L^{\infty}(\tilde{Q}_{\tau,T})\), \(u_{m}\in H^{1}_{0}(\mathcal {O}_{t})\cap L^{\infty}(\mathcal {O}_{t})\) for a.e. \(t\in(\tau,T)\), and \(\vert u_{m}(t)\vert ^{(\frac{N}{N-2})^{h+1}}\in H^{1}_{0}(\mathcal {O}_{t})\) for a.e. \(t\in(\tau,T)\), again with the embedding
to get
for any \(s-\tau\geqslant \tilde{T}_{h}(t,\hat{D})+1\). Therefore, we set
and
from (4.19) and (4.21) we know that (\(A_{h+1}\)) and (\(B_{h+1}\)) hold. □
4.2 Higher-order integrability for variational solutions
Now, as initial data \(u_{\tau}\in L^{2}(\mathcal {O}_{\tau})\), we establish the following results for the variational solution of equation (1.3).
Theorem 4.2
Let \(\hat{D}=\{D(\tau):\tau\in\mathbb{R}\}\in \mathscr{D}_{\lambda_{1}}\). Then for each \(t\in\mathbb{R}\) and each \(\delta\in[0,\infty)\), there exist constants \(T(t,\delta,\hat{D})\) and \(M_{\delta}(t)\) such that
Proof
For each fixed \(t\in\mathbb{R}\) and \(h\in\{0,1,2,\ldots\}\), take
where \(\tilde{T}_{h}(t,\hat{D})\) is just the constant given in Theorem 4.1 corresponding to the pair t, h.
Setting
and for any (fixed) τ satisfying
we consider the approximation on the interval \([\tau, T]\).
When \(u_{\tau}\in L^{2}(\mathcal {O}_{\tau})\), there exists a sequence \(\{u_{\tau m}\}\subset L^{\infty}(\mathcal {O}_{\tau}) \) such that
On account of the proof of Proposition 11 in [10], we have
for all \(t\in[\tau,T]\) and \(n,m\geqslant1\). By applying the Gronwall lemma, we conclude from the inequality above that
On the other hand, by (4.9), it can be proved that
Consequently, from (4.23) and (4.24), it follows that
in particular, for each \(t\in[\tau,T]\), there exists a subsequence \(\{ u_{m j}\}\) such that
Therefore, set \(\bar{M}_{h}(t)=\tilde{M}_{h}(t)\) for (\(A_{h}\)) and (\(B_{h}\)) in Theorem 4.1, and as \(m\to+\infty\), according to Fatou’s lemma:
for any \(t-\tau\geqslant T_{h}(t,\hat{D})\) holds.
For each \(\delta\in[0,\infty)\), there is a unique \(h\in\{1,2,3,\ldots\}\) such that
Then, for each \(\delta\in[0,\infty)\) and any\(t-\tau\geqslant T_{h}(t,\hat{D})\), we have
and let
(which depends only on δ, t, h, N, β, \(\Vert f(t)\Vert _{L^{\infty}(\mathcal {O}_{t})}\), \(\lambda_{1,t}\), and \(\vert \mathcal {O}_{t}\vert \)) and \(T(t,\delta,\hat{D}):=T_{h}(t,\hat{D})\) (where the constant h is fixed by (4.25)), we know that, for any \(t\in\mathbb{R}\), any \(\mathcal{D}(\tau)\in\mathscr{D}_{\lambda_{1}}\), there exist constants \(T(t,\delta,\hat{D})\) and \(M_{\delta}(t)\) such that
□
4.3 The proof of Theorem 1.2
Proof
It is sufficient to verify the conditions of Theorem 2.4. According to Theorem 2.6, the norm-to-weak continuity is easily checked.
For any \(t\in\mathbb{R}\), denote
then, according to Theorem 4.2, for any \(t\in\mathbb{R}\) and any \(u_{\tau}\in D(\tau)\), there exist \(\mathcal{B}(t)\) and \(T(t,\delta ,\hat{D})\) such that
for any \(t-\tau\geqslant T(t,\delta,\hat{D})\), that is, there exists a pullback \(\mathscr{D}_{\lambda_{1}}\) absorbing set in \(L^{2+\delta}\) (\(\forall\delta\in[0,+\infty)\)).
By Theorem 4.2, we know that, for any \(\varepsilon>0\), any \(\delta\in[0,+\infty)\), and each \(t\in \mathbb{R}\), there exist positive constants \(\bar{T}(t,\delta,\hat{D},\varepsilon)\) and M̄ such that
for any \(t-\tau\geqslant \bar{T}(t,\delta,\hat{D},\varepsilon)\). Combining Theorem 2.5 and Theorem 2.10, we prove that the process \(U(t,\tau)\) associated to (1.3) is pullback ω-\(\mathscr{D}\)-limit compact in \(L^{2+\delta}(\mathcal {O}_{t})\). □
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Acknowledgements
The author would like to thank the referees for their detailed suggestions, which helped to improve the original manuscript. This work was supported by Tian Yuan Fund of Mathematics 11326100, Natural Science Foundation of Gansu Province 145RJZA033 and Gansu Province science technology plan 1606RJZA041.
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Xiao, Y. Pullback attractors for non-autonomous reaction-diffusion equation in non-cylindrical domains. Adv Differ Equ 2016, 230 (2016). https://doi.org/10.1186/s13662-016-0888-1
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DOI: https://doi.org/10.1186/s13662-016-0888-1
MSC
- 35K57
- 35B40
- 35B41
Keywords
- variational solution
- pullback \(\mathscr{D}_{\lambda_{1}}\) attractor
- higher-order integrability
- non-cylindrical domains