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Dynamic of the nonclassical diffusion equation with memory
Boundary Value Problems volume 2023, Article number: 79 (2023)
Abstract
In this paper, we consider the nonclassical diffusion equation with memory and the nonlinearity of the polynomial growth condition of arbitrary order in the time-dependent space. First, the well-posedness of the solution for the equation is obtained in the time-dependent space \(\mathscr{U}_{t}\). Then, we establish the existence and regularity of the time-dependent global attractor. Finally, we also conclude that the fractal dimension of the time-dependent attractor is finite.
1 Introduction
In recent years, exploring the dynamic behavior of dissipative partial differential equations with time-dependent coefficients in the field of infinite-dimensional dynamical systems has attracted much attention. For this kind of problem, Plinio et al. [36] first put forward the concept of the time-dependent global attractor in time-dependent space. Subsequently, improved theoretical results and some applications have emerged widely, see [12, 14, 16, 24, 25, 27–34, 37, 43, 50]. The main characteristic of this type of problem is that the norm of space depends on time explicitly, which will lead to the fact that the considered problem is still nonautonomous even when the forcing term is independent of time t.
In this paper, we are concerned with the following nonclassical diffusion equation with memory
in time-dependent space, where \(\Omega \subset \mathbb{R}^{N}\) is a bounded smooth domain, \(u=u(x,t): \Omega \times [\tau ,\infty )\rightarrow \mathbb{R}\) is an unknown function, \(u_{\tau}(x,r): \Omega \times (-\infty ,\tau ]\) is the initial value function that characterizes the past time and \(g=g(x)\in H^{-1}(\Omega )\) is the external term. In addition, the time-dependent coefficient \(\varepsilon (t)\in C^{1}(\mathbb{R})\) is a decreasing bounded function satisfying
and there is a constant \(L>0\) such that
The nonlinearity \(f(s)\in C^{1}(\mathbb{R})\) with \(f(0)=0\) satisfies the polynomial growth condition of arbitrary order
and the dissipation condition
where \(\gamma _{i}\), \(\beta _{i}\) (\(i=1,2\)) and l are positive constants. Setting \(F(s)=\int _{0}^{s}f(y)\,dy\), it follows from (1.4) that there exist \(\tilde{\gamma _{i}}\), \(\tilde{\beta _{i}}\) (\(i=1,2\)) such that
The memory kernel κ is a nonnegative summable function satisfying \(\int _{0}^{\infty}\kappa (s)\,ds=1\) and having the following form
where \(\mu \in L^{1}(\mathbb{R}^{+})\) is a decreasing piecewise absolutely continuous function and is allowed to have infinitely many discontinuity points. We assume
From [17], the above inequality (1.8) is equivalent to the following
where \(M\geq 1\), \(\delta >0\), \(r\geq 0\).
The nonclassical diffusion equation arising from several physical phenomena, a pseudoparabolic equation, was first proposed by Aifantis in [1]. Then, Jäckle [19] came up with the diffusion equation with memory in the study of heat conduction and relaxation of high-viscosity liquids. Gradually, the study of nonclassical diffusion equations with memory emerged in the time-dependent spaces.
The researches were first focused on the nonclassical diffusion equation with constant coefficient. The kind of problem where the function \(\varepsilon (t)\) is a positive constant and the case \(\kappa (s)=0\) or \(\kappa (s)\neq 0\) in equation (1.1) has been studied extensively, see, e.g., [2–7, 9–11, 20–22, 26, 39, 41, 42, 44–48] and the references therein. For the nonclassical diffusion equation with memory (i.e., \(\kappa (s)\neq 0\) in (1.1)), the authors [46] first obtained the existence and regularity of a uniform attractor in \(H_{0}^{1}(\Omega )\times L_{\mu}^{2}(\mathbb{R}^{+},H_{0}^{1}( \Omega ))\) (\(\Omega \in \mathbb{R}^{N}\), \(N\geq 3\)), when the nonlinearity satisfies the critical exponential growth condition and the memory kernel satisfies
Since then, the condition (1.10) has been used for the nonclassical diffusion equation with memory and constant coefficient. In 2014, Conti et al. [11] applied the memory kernel condition (1.8) and proved the existence of global attractor in \(H_{0}^{1}(\Omega )\times L_{\mu}^{2}(\mathbb{R}^{+},H_{0}^{1}( \Omega ))\) (\(\Omega \in \mathbb{R}^{3}\)). Then, they also obtained the existence of the exponential attractor in [10]. In [2], the authors obtained the existence of a global attractor for the nonclassical diffusion equation with memory and a new class of nonlinearity. It is worth noting that (1.9) is weaker than (1.10), which shows (1.8) is more general, see [10, 11] for details.
When the perturbed coefficient \(\varepsilon (t)\) is a decreasing function and satisfies (1.2) and (1.3), there are some results for other equations, see [12, 24, 25, 27, 28, 30–32, 40]. For the nonclassical diffusion equation, the case \(\kappa (s)=0\) in equation (1.1) has been investigated by some authors. When the forcing term \(g\in L^{2}(\Omega )\) (\(\Omega \subset \mathbb{R}^{3}\)) and the nonlinearity \(f(u)\) satisfies \(\vert f'(u)\vert \leq C(1+\vert u\vert )\), the authors [16] obtained the existence of a time-dependent global attractor by using the decomposition technique. Using the same method, Ma et al. proved the existence, regularity, and asymptotic structure of the time-dependent global attractor in [29], when the forcing term \(g\in H^{-1}(\Omega )\) (\(\Omega \subset \mathbb{R}^{N}\), \(N\geq 3\)) and the nonlinear term \(f(u)\) satisfies the critical exponential growth condition. In [43, 50], in order to overcome the difficulty caused by the nonlinearity with a polynomial growth condition of arbitrary order, the authors applied the contractive function method and obtained the existence of the time-dependent global attractor. However, we also proved the regularity and asymptotic structure of the time-dependent global attractor in [43]. However, we find that all the articles mentioned above studied the nonclassical diffusion equations without memory. Recently, we submitted a new manuscript on the existence of the time-dependent global attractor for the nonclassical diffusion equation with memory and the nonlinearity of critical exponential growth. In summary, the researches of the nonclassical diffusion equations are not abundant on time-dependent spaces.
Therefore, these discussions above motivate us to consider the dynamic behavior of problem (1.1) in the time-dependent space in this paper. In contrast to the existing papers, the nonlinear term satisfies the polynomial growth condition of arbitrary order and the memory kernel condition is weaker in this paper, which is a highlight worth mentioning.
In order to obtain the corresponding results for problem (1.1), we follow the ideas from [11, 43]. However, we also encountered some difficulties. First, a weaker memory kernel condition (1.8) makes the directly obtained energy functional unavailable. Secondly, due to the influence of the term \(- \varepsilon (t)\triangle u_{t}\), the solution for the problem (1.1) does not have the higher regularity. Finally, we can not apply the decomposition technique from [29, 36], because of the particularity of the nonlinearity for a polynomial growth condition of arbitrary order. For these reasons, introducing a new function related to the memory kernel, we construct a new energy functional and obtain the existence of the time-dependent absorbing sets. Then, we apply the contractive function method rather than the decomposition method to verify the asymptotic compactness. In addition, we apply a decomposition method from [43] and obtain the regularity of the time-dependent attractor, and we also conclude that the fractal dimension of the time-dependent attractor is finite.
The paper is organized as follows. In Sect. 2, we introduce notations, function spaces involved, some abstract results for the time-dependent global attractor, and some standard conclusions. In Sect. 3, we will prove the well-posedness of the solution. Then, based on the existence of the solution, we obtain the process generated by a weak solution. In Sect. 4, we investigate the existence of the time-dependent global attractor in \(\mathscr{U}_{t}\). In Sect. 5, the regularity of the time-dependent attractor is obtained in \(\mathscr{U}^{1}_{t}\). Finally, based on the previous results in this paper, we find that the fractal dimension of the time-dependent attractor is finite in \(\mathscr{U}_{t}\).
2 Preliminaries
As in [15], we introduce a variable that shows the past history of equation (1.1), that is
and
where \(\eta _{t}=\frac{\partial \eta}{\partial t}\), \(\eta _{s}= \frac{\partial \eta}{\partial s}\).
Therefore, according to (1.7), (2.1), and (2.2), the problem (1.1) can be transformed into the following system
with the corresponding initial conditions
First, we give some spaces and the corresponding norms used in the remainder of the paper. Usually, we set \(\Vert \cdot \Vert _{L^{p}(\Omega )}\) as the norm of \(L^{p}(\Omega )\) (\(p\geq 1\)). In particular, set \(\langle \cdot ,\cdot \rangle \) and \(\Vert \cdot \Vert \) as the scalar product and norm of \(\mathrm{H}=L^{2}(\Omega )\), respectively. The Laplacian \(A=-\triangle \) with Dirichlet boundary conditions is a positive operator on H with domain \(H^{2}(\Omega )\cap H_{0}^{1}(\Omega )\). Then, we consider the family of Hilbert spaces \(\mathrm{H}_{s}=\mathrm{D}(A^{s/2})\), \(\forall s\in \mathbb{R}\), with the standard inner products and norms, respectively,
In particular, \(\mathrm{H}_{-1}=H^{-1}(\Omega )\), \(\mathrm{H}_{0}=\mathrm{H}\), \(\mathrm{H}_{1}=H_{0}^{1}(\Omega )\), \(\mathrm{H}_{2}=H^{2}(\Omega )\cap H_{0}^{1}(\Omega )\).
Hence, for any \(t\in \mathbb{R}\), \(-1\leq s\leq 1\), we define the time-dependent spaces \(\mathcal{H}_{t}^{s}\) with norms
and
where the embedding constant is independent of \(t\in \mathbb{R}\) and the symbol s is always omitted whenever zero. In particular,
According to the definition of the memory kernel, for any \(-1\leq s\leq 1\), we introduce the Hilbert (history) spaces
with the corresponding inner products and norms
We know that \(\mathcal{M}^{s_{1}}\hookrightarrow \mathcal{M}^{s_{2}}\), \(\forall s_{1}>s_{2}\). Due to the lack of tightness in the memory space, a new weighted space needs to be constructed. According to the literatures [13], let
and
where \(T_{\eta ^{t}}\) is the tail function of \(\eta ^{t}\) with the following form
Now, combining the above spaces, we give the following time-dependent space families
endowed with the inner products and norms, respectively,
In particular,
Note that the dual space of X is denoted as \(X^{*}\). As a convenience, we choose C as the positive constant depending on the subscript that may be different from line to line or in the same line throughout the paper.
Secondly, we recall some notations, concepts, and abstract results, see, e.g., [14, 33, 36] for more details. For every \(t\in \mathbb{R}\), let \(X_{t}\) be a family of normed spaces, we introduce the R-ball of \(X_{t}\)
In addition, we denote the Hausdorff semidistance of two nonempty sets \(B, C \subset X_{t}\) by
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 \(\{U(t, \tau ):X_{\tau}\rightarrow X_{t}, t\geq \tau \in \mathbb{R}\}\) with properties
-
(i)
\(U(\tau ,\tau )=\mathrm{Id}\) is the identity on \(X_{\tau}\), \(\tau \in \mathbb{R}\);
-
(ii)
\(U(t,s)U(s,\tau )=U(t,\tau )\), \(\forall t\geq s \geq \tau \).
Definition 2.2
A family \(\mathfrak{D}=\{D_{t}\}_{t\in \mathbb{R}}\) of bounded sets \(D_{t}\subset X_{t}\) is called uniformly bounded if there exist a constant \(R>0\) such that \(D_{t}\subset \mathbb{B}_{X_{t}}(R)\), \(\forall t\in \mathbb{R}\).
Definition 2.3
A time-dependent absorbing set for the process \(\{U(t,\tau )\}_{t\geq \tau}\) is a uniformly bounded family \(\mathfrak{B}=\{B_{t}\}_{t\in \mathbb{R}}\) with the following property: for every \(R>0\) there exists a \(t_{0}\) such that
Definition 2.4
We say that a process \(\{U(t,\tau )\}_{t\geq \tau}\) in a family of normed spaces \(\{X_{t}\}_{t\in \mathbb{R}}\) is pullback asymptotically compact if and only if for any fixed \(t\in \mathbb{R}\), bounded sequence \(\{x_{n}\}_{n=1}^{\infty}\subset X_{\tau _{n}}\) and any \(\{\tau _{n}\}_{n=1}^{\infty}\subset \mathbb{R}^{-t}\) with \(\tau _{n}\rightarrow -\infty \) as \(n\rightarrow \infty \), sequence \(\{U(t,\tau _{n})x_{n}\}_{n=1}^{\infty}\) has a convergent subsequence, where \(\mathbb{R}^{-t}=\{\tau : \tau \in \mathbb{R}, \tau \leq t\}\).
Definition 2.5
The time-dependent global attractor for \(\{U(t,\tau )\}_{t\geq \tau}\) is the smallest family \(\mathfrak{A}=\{A_{t}\}_{t\in \mathbb{R}}\) such that
-
(i)
each \(A_{t}\) is compact in \(X_{t}\);
-
(ii)
\(\mathfrak{A}\) is pullback attracting, i.e., it is uniformly bounded and the limit
$$ \lim_{\tau \rightarrow -\infty}\operatorname{dist}_{X_{t}}\bigl(U(t,\tau )D_{\tau},A_{t}\bigr)=0 $$
holds for every uniformly bounded family \(\mathfrak{D}=\{D_{t}\}_{t\in \mathbb{R}}\) and every fixed \(t\in \mathbb{R}\).
Definition 2.6
We say \(\mathfrak{A}=\{A_{t}\}_{t\in \mathbb{R}}\) is invariant if
Definition 2.7
Let \(\{X_{t}\}_{t\in \mathbb{R}}\) be a family of Banach spaces and \(\mathfrak{C}=\{C_{t}\}_{t\in \mathbb{R}}\) be a family of uniformly bounded subsets of \(\{X_{t}\}_{t\in \mathbb{R}}\). We call a function \(\psi _{\tau}^{t}(\cdot ,\cdot )\), defined on \(X_{t}\times X_{t}\), a contractive function on \(C_{\tau}\times C_{\tau}\) if for any fixed \(t\in \mathbb{R}\) and any sequence \(\{x_{n}\}_{n=1}^{\infty}\subset C_{\tau}\), there is a subsequence \(\{x_{n_{k}}\}_{n=1}^{\infty}\subset \{x_{n}\}_{n=1}^{\infty}\) such that
Theorem 2.8
Let \(\{U(t,\tau )\}_{t\geq \tau}\) be a process \(\{X_{t}\}_{t\in \mathbb{R}}\) and has a pullback absorbing family \(\mathfrak{B}=\{B_{t}\}_{t\in \mathbb{R}}\). Moreover, assume that for any \(\epsilon >0\) there exists \(T=T(\epsilon )\leq t\), \(\psi _{T}^{t}\in \mathfrak{C}(B_{T})\) such that
for any fixed \(t\in \mathbb{R}\). Then \(\{U(t,\tau )\}_{t\geq \tau}\) is pullback asymptotically compact.
Theorem 2.9
Let \(\{U(t,\tau )\}_{t\geq \tau}\) be a process in a family of Banach spaces \(\{X_{t}\}_{t\in \mathbb{R}}\). Then, \(U(\cdot ,\cdot )\) has a time-dependent global attractor \(\mathcal{A}=\{A_{t}\}_{t\in \mathbb{R}}\) satisfying \(A_{t}= \bigcap_{s\leq t} \overline{\bigcup_{\tau \leq s}U(t,\tau )B_{\tau}}\) if and only if
-
(i)
\(\{U(t,\tau )\}_{t\geq \tau}\) has a pullback absorbing family \(\mathfrak{B}=\{B_{t}\}_{t\in \mathbb{R}}\);
-
(ii)
\(\{U(t,\tau )\}_{t\geq \tau}\) is pullback asymptotically compact.
For the sake of estimation, we next recall some standard conclusions, see [18, 23, 35].
Lemma 2.10
Assume that the memory function κ satisfies (1.7) and (1.8), then for any \(T>\tau \), \(\eta ^{t}\in C([\tau , T],L_{ \mu}^{2}(\mathbb{R}^{+};\mathrm{H}_{1}))\) such that
Lemma 2.11
If \(\mathcal{K}\subset \mathcal{M}\) satisfies
-
(i)
\(\sup_{\eta ^{t}\in \mathcal{K}}\Vert \eta ^{t}\Vert _{\mathcal{M}^{1}}< \infty \);
-
(ii)
\(\sup_{\eta ^{t}\in \mathcal{K}}\Vert \eta ^{t}_{s}\Vert _{\mathcal{M}}< \infty \);
-
(iii)
\(\lim_{y\rightarrow \infty}[\sup_{\eta ^{t}\in \mathcal{K}}T_{ \eta ^{t}}(y)]=0\),
then \(\mathcal{K}\) is relatively compact in \(\mathcal{M}\).
Lemma 2.12
(Aubin–Lions Lemma). Assume that X, B, and Y are three Banach spaces with \(X\hookrightarrow \hookrightarrow B\) and \(B\hookrightarrow Y\). Let \(f_{n}\) be bounded in \(L^{p}([0,T],B)\) (\(1\leq p<\infty \)). If \(f_{n}\) satisfies
-
(i)
\(f_{n}\) is bounded in \(L^{p}([0,T],X)\);
-
(ii)
\(\frac{\partial f_{n}}{\partial t}\) is bounded in \(L^{p}([0,T],Y)\),
then \(f_{n}\) is relatively compact in \(L^{p}([0,T],B)\).
3 Well-posedness
Now, we give the definition of a weak solution and prove the well-posedness of the weak solution for the problem (2.3) and (2.4) by using the Faedo–Galerkin method from [8, 23, 38].
Definition 3.1
The function \(z=(u,\eta ^{t})=(u(x,t),\eta ^{t}(x,s))\) defined in \(\Omega \times [\tau ,T]\) is said to be a weak solution for the problem (2.3) and (2.4) with the initial data \(z_{\tau}\in \mathbb{B}_{\mathscr{U}_{\tau}}(R_{0})\subset \mathscr{U}_{\tau}\), \(-\infty <\tau <T<+\infty \) if z satisfies
-
(i)
\(z\in C([\tau ,T],\mathscr{U}_{t})\), \((x,t)\in \Omega \times [\tau ,T]\);
-
(ii)
for any \(\theta =(v,\xi ^{t})\in \mathscr{U}_{t}\), the equality
$$ \langle u_{t},v\rangle +\varepsilon (t)\langle \nabla u_{t},\nabla v \rangle +\langle \nabla u,\nabla v\rangle +\bigl\langle \eta ^{t},v \bigr\rangle _{\mu ,1}+\bigl\langle f(u), v\bigr\rangle =\langle g,v\rangle $$and
$$ \bigl\langle \eta ^{t}_{t},\xi ^{t}\bigr\rangle _{\mu ,1}=-\bigl\langle \eta _{s}^{t}, \xi ^{t}\bigr\rangle _{\mu ,1}+\bigl\langle u,\xi ^{t}\bigr\rangle _{\mu ,1} $$hold for a.e. \([\tau ,T]\).
Theorem 3.2
Assume that (1.2)–(1.8) hold and \(g\in H^{-1}(\Omega )\), then for any initial data \(z_{\tau}=(u_{\tau},\eta ^{\tau})\in \mathbb{B}_{\mathscr{U}_{\tau}}(R_{0}) \subset \mathscr{U}_{\tau}\) and any \(\tau \in \mathbb{R}\), there exists a unique solution z for the problem (2.3) and (2.4) such that \(z=(u,\eta ^{t})\in C([\tau ,T],\mathscr{U}_{t})\) for any fixed \(T>\tau \). Furthermore, the solution depends on the initial data continuously in \(\mathscr{U}_{t}\).
Proof
Assume that \(\omega _{k}\) is the eigenfunction of \(A=- \triangle \) with a Dirichlet boundary value in \(\mathrm{H}_{1}\), then \(\{\omega _{k}\}_{k=1}^{\infty}\) is a standard orthogonal basis of H and is also an orthogonal basis in \(\mathrm{H}_{1}\). The corresponding eigenvalues are denoted by \(0<\lambda _{1}\leq \lambda _{2}\leq \cdots \leq \lambda _{j}\leq \cdots \) , \(\lambda _{j}\rightarrow \infty \) with \(A\omega _{k}=\lambda _{k}\omega _{k}\), \(\forall k\in \mathbb{N}\). Next, we will complete our proof.
♡ Faedo–Galerkin scheme. If we give an integer m, we denote by \(P_{m}\) the projection on the subspace \(\operatorname{span}\{\omega _{1},\ldots ,\omega _{m}\}\) in \(H_{0}^{1}(\Omega )\), and \(Q_{m}\) the projection on the subspace \(\operatorname{span}\{e_{1},\ldots ,e_{m}\} \subset L^{2}_{\mu}(\mathbb{R}^{+},\mathrm{H}_{1})\) in \(L^{2}_{\mu}(\mathbb{R}^{+},\mathrm{H}_{1})\). For every fixed m, we look for the function \(u^{m}(t)=P_{m}u=\sum_{k=1}^{m}a_{m}^{k}(t)\omega _{k}\) and \(\eta ^{t,m}(s)=Q_{m}\eta ^{t}=\sum_{k=1}^{m}b_{m}^{k}(t)e_{k}(s)\), where \(a_{m}^{k}(t)\) and \(b_{m}^{k}(t)\) satisfy
On account of the standard existence theory for ordinary differential equations, there exists a continuous solution of the problem (2.3) and (2.4) on an interval \([\tau ,T]\). Then, we will prove the convergence of \(z^{m}(t)=(u^{m},\eta ^{t,m})\).
♡ Energy estimates. Multiplying the first and the second equation of (3.1) by \(a_{m}^{k}\) and \(b_{m}^{k}\), respectively, and summing from 1 to m about k, we have
By (1.4), Hölder’s inequality, and Young’s inequality, we obtain
According to Lemma 2.10 and (3.2)–(3.4), we have
Applying the decreasing property of \(\varepsilon (t)\) and integrating from Ï„ to t at the sides of (3.5) we obtain
where
Thereby, we infer from (3.6) that
for any fixed \(T>t\). It follows from (1.4) that
where \(\frac{1}{p}+\frac{1}{q}=1\). Hence, we infer from (3.9) that
Next, we verify the uniform estimate for \(u_{t}^{m}\). Multiplying the first equation of (3.1) by \(\partial _{t}a_{m}^{k}\) and summing from 1 to m yields
where
Applying (1.6) arrives at
Hence, combining with Poincaré’s inequality, (3.4), (3.12), and (3.13), we obtain
In addition,
Then, it follows from (3.6), (3.11), and (3.16) that
for \(t\in [\tau ,T]\). Integrating from s to t at the sides of (3.17), for any \(s\in (\tau ,T]\), we obtain
Then, integrating from Ï„ to T about variable s for (3.18), we have
From (3.6), (3.14), (3.15), and (3.19), then there exists \(\rho _{1}>0\) such that
where
Combining with (3.14), (3.15), and (3.20) as well as integrating from Ï„ to t at the side of (3.17), we obtain
where
Hence, we infer from (3.21) that
♡ Existence of solution. It follows from (3.7), (3.8), (3.10), and (3.22) that there exist \(u\in L^{\infty}([\tau ,T],\mathcal{H}_{t})\cap L^{2}([\tau ,T], \mathrm{H}_{1})\cap L^{2}([\tau ,T],L^{p}(\Omega ))\), \(\eta ^{t}\in L^{\infty}([\tau ,T], L_{\mu}^{2}(\mathbb{R}^{+}, \mathrm{H}_{1}))\), \(\chi \in L^{q}([\tau ,T],L^{q}(\Omega ))\), \(u_{t} \in L^{2}([\tau ,T],\mathcal{H}_{t})\) and a subsequence of \(\{u^{m}\}_{m=1}^{\infty}\) (still denoted as \(\{u^{m}\}_{m=1}^{\infty}\)) such that
We find from (3.6), (3.21), and Lemma 2.12 that there is a subsequence of \(\{u^{m}\}_{m=1}^{\infty}\) (still denoted as \(\{u^{m}\}_{m=1}^{\infty}\)) such that
which shows that
Due to (3.29) and the continuity of f, we have
which combines with the term-by-term of the integral theorem of Lebesgue and the uniqueness of limit, we verify \(\chi =f(u)\).
Moreover, we can obtain that \(z\in C([\tau ,T],\mathscr{U}_{t})\), the conclusion (ii) in Definition 3.1 and \(z(\tau )=z_{\tau}\) hold. The proof of these conclusions is similar to Theorem 3.2 from [43] and also trivial, hence, we will omit it.
♡ Uniqueness and continuity of solution. Assume that \((u^{i},\eta ^{t,i})\) (\(i=1, 2\)) are two solutions of the problem (2.3) and (2.4) with the initial data \((u_{\tau}^{i},\eta ^{\tau ,i})\), respectively. For convenience, define \(\bar{u}=u^{1}-u^{2}\), \(\bar{\eta}^{t}=\eta ^{t,1}- \eta ^{t,2}\), then \((\bar{u},\bar{\eta}^{t})\) satisfies the following problem
Multiplying the first equation (3.30) by ū and integrating on Ω, we obtain
In view of (1.2), (1.3), (1.5), and Lemma 2.10, then
Then, by the Gronwall lemma, we obtain
which shows the uniqueness and continuous dependence of the solution on the initial value. □
According to Theorem 3.2, we can define a continuous process \(\{U(t,\tau )\}_{t\geq \tau}\) generated by the solution of the problem (2.3) and (2.4), where the mapping
and \(U(t,\tau )z_{\tau}=z(t)\), \(z_{\tau}\in \mathscr{U}_{\tau}\).
4 The time-dependent global attractor
In this subsection, we first consider a time-dependent absorbing family for the solution process to prove the existence of the time-dependent global attractor.
Theorem 4.1
Assume that (1.2)–(1.4), (1.6)–(1.8), \(g \in H^{-1}(\Omega )\) hold and \(z_{\tau}=(u_{\tau},\eta ^{\tau})\in \mathbb{B}_{\mathscr{U}_{\tau}}(R_{0}) \subset \mathscr{U}_{\tau}\), then there exists \(R_{1}>0\) such that \(\mathfrak{B}=\{B_{t}\}_{t\in \mathbb{R}}=\{\mathbb{B}_{\mathscr{U}_{t}}(R_{1}) \}_{t\in \mathbb{R}}\) is a time-dependent absorbing set in \(\mathscr{U}_{t}\) for the process \(\{U(t,\tau )\}_{t\geq \tau}\) corresponding to the problem (2.3) and (2.4).
Proof
Multiplying (2.3) by z and repeating the estimates of Theorem 3.2, we can obtain
where
According to (1.2), (1.3), (4.1), and Poincaré’s inequality, we have
To reconstruct \(E_{1}(t)\), we assume a new function
We find from (1.8) that
In addition, taking the derivative with respect to t at the side of (4.3) and combining with (1.7) and (1.8), we obtain
Therefore, for fixed \(\nu >0\), we definite the function
It follows from (4.2), (4.5), and (4.6) that
Let \(\sigma _{1}=\min \{\frac{1}{2L},\frac{\lambda _{1}}{8}, \frac{\nu}{32\Theta ^{2}\kappa (0)}\}>0\), then
for small enough ν. By (4.6), we also yield
It follows from (4.7) and (4.8) that
By the Gronwall lemma, we obtain
Hence, from (4.8) and (4.10), we conclude that
that is,
for any \(t\geq t^{\ast}=\tau +\frac{1}{\sigma _{1}}\ln \frac{4E_{1}(\tau )}{R_{1}}\), where \(R_{1}=\frac{4}{\sigma _{1}}(\Vert g\Vert ^{2}_{-1}+\beta _{1}\vert \Omega \vert )\).
Therefore, \(B_{t}=\{z=(u,\eta ^{t})\in \mathscr{U}_{t}:\Vert z(t)\Vert ^{2}_{\mathscr{U}_{t}} \leq R_{1}\}\) is a time-dependent absorbing set in \(\mathscr{U}_{t}\) for the solution process \(\{U(t,\tau )\}_{t\geq \tau}\). □
We next verify the pullback asymptotically compact for the process \(\{U(t,\tau )\}_{t\geq \tau}\) corresponding to the problem (2.3) and (2.4).
Theorem 4.2
Assume that (1.2), (1.3), (1.4), and (1.8) hold, then the process \(\{U(t,\tau )\}_{t\geq \tau}\) of the problem (2.3) and (2.4) is pullback asymptotically compact in \(\mathscr{U}_{t}\).
Proof
Assume that \(z^{n}=(u^{n},\eta ^{t,n})\), \(z^{m}=(u^{m},\eta ^{t,m})\) are two solutions of the problem (2.3) and (2.4) with initial data \(z^{n}_{\tau}, z^{m}_{\tau}\in \mathbb{B}_{\mathscr{U}_{\tau}}(R_{0})\), respectively. Without loss of generality, we assume \(\tau \leq T_{1}< t\) for every fixed \(T_{1}\). As a convenience, let \(w(t)=u^{n}(t)-u^{m}(t)\), \(\zeta ^{t}=\eta ^{t,n}-\eta ^{t,m}\), then \((w(t),\zeta ^{t})\) satisfies the following system
Taking w as a test function for the first equation of (4.11), we can obtain
where
Combining with (1.2), (1.3), and Poincaré’s inequality, we have
Let
then applying the similar arguments used in the proof of Theorem 4.1, we find that
where \(0<\sigma _{2}=\min \{\frac{1}{2L},\frac{\lambda _{1}}{4}, \frac{\nu}{16\Theta ^{2}\kappa (0)}\}\), \(\nu >0\) is small enough. It follows from (4.14) and (4.15) that
By the Gronwall lemma, we obtain
Hence, for any \(\epsilon >0\) and some given t, set \(t>T_{1}\geq \tau \) such that \(t-T_{1}\) is enough large, we can conclude from (4.15) and (4.17) that
where
Now, assume that \(z^{k}=(u^{k},\eta ^{t,k})\) is a solution of the problem (2.3) and (2.4) with initial data \(z^{k}_{\tau}\in \mathbb{B}_{\mathscr{U}_{\tau}}(R_{0})\), then we find that \(u_{t}^{k}\in L^{2}([T_{1},t],\mathcal{H}_{t})\) and \(u^{k}\in L^{2}([T_{1},t],H_{0}^{1}(\Omega ))\) for some given t by applying the same arguments of Theorem 3.2. Thereby, we infer from Lemma 2.12 that there exists a convergent subsequence of \(u^{k}\) (denoted as \(u^{k_{i}}\)) such that
which implies that \(\psi _{T_{1}}^{t}\in \mathfrak{C}(B_{T_{1}})\). We conclude from (4.18), (4.19), and Theorem 2.8 that
This shows that the process \(\{U(t,\tau )\}_{t\geq \tau}\) is pullback asymptotic compact in \(\mathscr{U}_{t}\). □
Theorem 4.3
The process \(\{U(t,\tau )\}_{t\geq \tau}\) generated by the problem (2.3) and (2.4) has an invariant time-dependent global attractor \(\mathfrak{A}=\{A_{t}\}_{t\in \mathbb{R}}\) in \(\mathscr{U}_{t}\).
Proof
Combining with Theorem 4.1 and Theorem 4.2, we obtain easily the existence of the invariant time-dependent global attractor \(\mathfrak{A}=\{A_{t}\}_{t\in \mathbb{R}}\) for the problem (2.3) and (2.4). □
5 Regularity of the attractor
In this subsection, based on the ideas from [49], we obtain the uniform boundedness (i.e., regularity) of the attractor. Now, we resolve the solution \(U(t,\tau )z_{\tau}=z(t)=(u(t),\eta ^{t})\) with \(z_{\tau}\in A_{\tau}\) into the sum
where \(U_{0}(t,\tau )z_{\tau}=(v(t),\xi ^{t})\), \(U_{1}(t,\tau )z_{\tau}=(y(t), \zeta ^{t})\) solve the following system, respectively,
and
Note that for every \(g\in H^{-1}(\Omega )\) and any \(\vartheta >0\), there exists a \(g^{0}\in L^{2}(\Omega )\) such that
as \(L^{2}(\Omega )\hookrightarrow H^{-1}(\Omega )\) is dense.
Lemma 5.1
Assume that (1.2)–(1.8) hold and \(g\in H^{-1}(\Omega )\), then
where \(\sigma _{1}\) is given in Theorem 4.1.
Proof
Similar to the proof of Theorem 4.1, we find easily that (5.4) holds. Therefore, the proof is omitted. □
Lemma 5.2
Assume that (1.2)–(1.8) hold and \(g^{0}\in L^{2}(\Omega )\), then there exists \(R_{2}>0\) such that
Proof
Taking the inner product of \(-\triangle y\) with the first equation of (5.2) in \(L^{2}(\Omega )\), we obtain
From (1.5) and Young’s inequality, then
Similar to Lemma 2.10, we also obtain
By (1.2), (1.3), and (5.5)–(5.8), we have
Set
then it is easy to obtain that
by using the same discussion of Theorem 4.1. Further, we have
where \(\sigma _{1}\) is given in Theorem 4.1,
 □
Theorem 5.3
Assume that (1.2)–(1.8) hold and \(g\in H^{-1}(\Omega )\), then \(\{A_{t}\}_{t\in \mathbb{R}}\) is bounded in \(\mathscr{U}_{t}^{1}\).
Proof
Thanks to Lemma 5.1 and Lemma 5.2, then for any \(t\in \mathbb{R}\), we have
where \(\sigma _{3}>0\),
Thereby, the above conclusion holds. □
6 Fractal dimension of the attractor
In what follows, in order to study the attractor further, we discuss the fractal dimension of the time-dependent global attractor by Lemma 6.1 of [36].
We decompose the solution \(U(t,\tau )z_{\tau}=z(t)=(u,\eta ^{t})\) with initial value \(z_{\tau}=(u_{\tau}, \eta ^{\tau})\in \mathscr{U}_{\tau}\) into the sum
then the corresponding solutions are \(U(t,\tau )z^{i}_{\tau}=z^{i}(t)\) for any initial value \(z^{i}_{\tau}\in \mathscr{U}_{\tau}\) (\(i=1,2\)), respectively. Now, we split the solution \(z^{1}-z^{2}\) with initial value \(z^{1}_{\tau}-z^{2}_{\tau}\) into the following form
where \((\tilde{v},\tilde{\zeta}^{t})=D(t,\tau )z^{1}_{\tau}-D(t,\tau )z^{2}_{ \tau}\) and \((\tilde{w},\tilde{\xi}^{t})=K(t,\tau )z^{1}_{\tau}-K(t,\tau )z^{2}_{ \tau}\) are the solutions of the following problem, respectively,
and
Lemma 6.1
Assume that the process \(\{U(t,\tau )\}_{t\geq \tau}\) is broken down into
then for every \(\tau \in \mathbb{R}\), \(z^{1}_{\tau}, z^{2}_{\tau}\in A_{\tau}\) there exists \(t_{\star}>0\) such that
where \(Q_{t_{\star}}>0\) only depends on \(t_{\star}\) and \(0\leq \rho <\frac{1}{4}\).
Proof
Taking the inner product of á¹½ with the first equation of (6.1) in \(L^{2}(\Omega )\), we arrive at
Similar to the proof of Theorem 4.2, we infer that
where \(\sigma _{2}\) is given in Theorem 4.2. That is,
For any \(z^{1}_{\tau}, z^{2}_{\tau}\in A_{\tau}\), choose \(t_{\star}=\frac{3\ln 2}{\sigma _{2}}>0\) such that
Thereby, (6.3) holds.
Taking the inner product of \(-\triangle \tilde{w}\) with the first equation of (6.2) in \(L^{2}(\Omega )\), we have
Combining with (1.3), (6.5), and the continuous dependence of the solution on the initial value, we obtain
Integrating both sides of (6.6) from Ï„ to t, we see that
for any \(z^{i}_{\tau}\in \mathbb{B}_{\mathscr{U}_{\tau}}(R_{0})\). Hence,
Thanks to Lemma 3.3 and Lemma 3.4 from [13], then
where C is a positive constant. By (6.8) and (6.9), we have
It follows from (6.10) that for any \(z^{1}_{\tau}, z^{2}_{\tau}\in A_{\tau}\) there exists \(t_{\star}>0\) such that
where \(Q_{\star}=Cle^{2lt_{\star}}\). Hence, (6.11) holds. □
Theorem 6.2
Assume that (1.2), (1.3), (1.5), (1.7), and (1.8) hold, then the time-dependent global attractor \(\mathfrak{A}=\{A_{t}\}_{t\in \mathbb{R}}\) has a finite fractal dimension in \(\mathscr{U}_{t}\), i.e.,
where
\(N_{q}(X,Y)\) is the minimum number of the q-ball of Y that covers K.
Proof
Repeating the similar proof of Lemma 5.2 for (6.5), we can obtain that \(\Vert \tilde{\xi}^{t}\Vert ^{2}_{\mu ,1}\) is bounded. Then, combining with (6.9) and Lemma 2.11, we find that \(\mathcal{L}^{2}\hookrightarrow \hookrightarrow \mathcal{M}^{1}\). Hence, \(\mathscr{Z}^{1}_{\tau +t_{\star}}\hookrightarrow \hookrightarrow \mathscr{U}_{\tau +t_{\star}}\). Then, we conclude from Lemma 6.1 ([36]) that
where \(\kappa _{q}=\sup_{t\geq 0}N_{q}(\mathbb{B}_{\mathscr{Z}^{1}_{t}}(1),Y)< \infty \). □
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References
Aifantis, E.C.: On the problem of diffusion in solids. Acta Mech. 37, 265–296 (1980)
Anh, C.T., Thanh, D.T.P., Toan, N.D.: Global attractors for nonclassical diffusion euqations with hereditary memory and a new class of nonlinearities. Ann. Pol. Math. 119, 1–21 (2017)
Anh, C.T., Thanh, D.T.P., Toan, N.D.: Averaging of nonclasssical diffusion equations with memory and singularly oscillating forces. Z. Anal. Anwend. 37, 299–314 (2018)
Anh, C.T., Toan, N.D.: Nonclassical diffusion equations on \(\mathbb{R}^{N}\) with singularly oscillating external forces. Appl. Math. Lett. 38, 20–26 (2014)
Cao, Y., Yin, J., Wang, C.: Cauchy problems of semilinear pseudo-parabolic equations. J. Differ. Equ. 246, 4568–4590 (2009)
Caraballo, T., Marquez-Durán, A.M., Rivero, F.: Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete Contin. Dyn. Syst., Ser. B 22, 1817–1833 (2017)
Chen, T., Chen, Z., Tang, Y.: Finite dimensionality of global attractors for a non-classion reaction-diffusion equation with memory. Appl. Math. Lett. 25, 357–362 (2012)
Chpyzhov, V.V., Vishik, M.I.: Attractor for Equations of Mathematical Physics. Amer Mathematical Society, Providence (2002)
Conti, M., Dell’Oro, F., Pata, V.: Nonclassical diffusion equations with memory lacking instantaneous damping. Commun. Pure Appl. Anal. 19, 2035–2050 (2020)
Conti, M., Marchini, E.M.: A remark on nonclassical diffusion equations with memory. Appl. Math. Optim. 73, 1–21 (2016)
Conti, M., Marchini, E.M., Pata, V.: Nonclassical diffusion with memory. Math. Methods Appl. Sci. 38, 948–958 (2014)
Conti, M., Pata, V.: On the time-dependent Cattaneo law in space dimension one. Appl. Math. Comput. 259, 32–44 (2015)
Conti, M., Pata, V., Squassina, M.: Singular limit of differential systems with memory. Indiana Univ. Math. J. 55, 169–215 (2006)
Conti, M., Pata, V., Temam, R.: Attractors for the process on time-dependent spaces. Applications to wave equation. J. Differ. Equ. 255, 1254–1277 (2013)
Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)
Ding, T., Liu, Y.: Time-dependent global attractor for the nonclassical diffusion equations. Appl. Anal. 94, 1439–1449 (2015)
Gatti, S., Miranville, A., Pata, V., et al.: Attractors for semilinear equations of viscoelasticity with very low dissipation. Rocky Mt. J. Math. 38, 1117–1138 (2008)
Grasselli, M., Pata, V.: Uniform attractors of non-autonomous systems with memory. Prog. Nonlinear Differ. Equ. Appl. 50, 155–178 (2002)
Jäckle, J.: Heat conduction and relaxation in liquids of high viscosity. Physica A 162, 377–404 (1990)
Kuttler, K., Aifantis, E.C.: Existence and uniqueness in nonclassical diffusion. Q. Appl. Math. 45, 549–560 (1987)
Le, T.T., Nguyen, D.T.: Uniform attractors of nonclassical diffusion equations on \(\mathbb{R^{N}}\) with memory and singularly oscillating external forces. Math. Methods Appl. Sci. 44, 820–852 (2021)
Lee, J., Toi, V.M.: Attractors for nonclassical diffusion equations with dynamic boundary conditions. Nonlinear Anal. 195, 111737 (2020)
Lions, J.L.: Quelques méthodes de Résolutions Des Problèms Aus Limites Nonlinéaries. Dunod Gauthier-Villars, Paris (1969)
Liu, T., Ma, Q.: Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete Contin. Dyn. Syst., Ser. B 23, 4595–4616 (2018)
Liu, T., Ma, Q.: Time-dependent attractor for plate equations on \(R^{n}\). J. Math. Anal. Appl. 479, 315–332 (2019)
Ma, Q., Liu, Y., Zhang, F.: Global attractors in \(H_{1}( \mathbb{R}^{N})\) for nonclassical diffusion equation. Discrete Dyn. Nat. Soc. 2012, Article ID 672762 (2012)
Ma, Q., Wang, J., Liu, T.: Time-dependent asymptotic behavior of the solution for wave equations with linear memory. Comput. Math. Appl. 76, 1372–1387 (2018)
Ma, Q., Wang, J., Liu, T.: Time-dependent attractor of wave equations with nonlinear damping and linear memory. Open Math. 17, 89–103 (2019)
Ma, Q., Wang, X., Xu, L.: Existence and regularity of time-dependent global attractors for the nonclassical reaction-diffusion equations with lower forcing term. Bound. Value Probl. 2016, 10 (2016)
Meng, F., Liu, C.: Necessary and sufficient condition for the existence of time-dependent global attractor and application. J. Math. Phys. 58, 1–9 (2017)
Meng, F., Wang, R., Zhao, C.: Attractor for a model of existensible beam with damping of time-dependent space. Topol. Methods Nonlinear Anal. 57, 365–393 (2021)
Meng, F., Wu, J., Zhao, C.: Time-dependent global attractor for extensible Berger equation. J. Math. Anal. Appl. 469, 1045–1069 (2019)
Meng, F., Yang, M., Zhong, C.: Attractors for wave equations with nonlinear damping on time-dependent on time-dependent space. Discrete Contin. Dyn. Syst., Ser. B 21, 205–225 (2015)
Pata, V., Conti, M.: Asymptotic structure of the attractor for process on time-dependent spaces. Nonlinear Anal., Real World Appl. 19, 1–10 (2014)
Pata, V., Zucchi, A.: Attractors for a damped hyperbolic equation with linear memory. Adv. Math. Sci. Appl. 11, 505–529 (2001)
Plinio, F.D., Duan, G., Temam, R.: Time dependent attractor for the oscillon equation. Discrete Dyn. Nat. Soc. 29, 141–167 (2011)
Plinio, F.D., Duan, G., Temam, R.: The 3-dimensional oscillon equation. Math. Phys. 1, 1–28 (2013)
Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press (2011)
Sun, C., Yang, M.: Dynamics of the nonclassical diffusion equations. Asymptot. Anal. 59, 51–81 (2008)
Sun, Y., Yang, Z.: Long time dynamics for a nonlinear viscoelastic equation with time-dependent memory kernel. Nonlinear Anal., Real World Appl. 64, 1–26 (2022)
Thanh, D.T.P., Toan, N.D.: Existence and long-time behavior of solutions to a class of nonclassical diffusion equations with infinite delays. Vietnam J. Math. 47, 309–325 (2019)
Toan, N.D.: Optimal control of nonclassical diffusion equations with memory. Acta Appl. Math. 169, 533–558 (2020)
Wang, J., Ma, Q.: Asymptotic dynamic of the nonclassical diffusion equation with time-dependent coefficient. J. Appl. Anal. Comput. 11, 445–463 (2021)
Wang, S., Li, D., Zhong, C.: On the dynamics of a class of nonclassical parabolic equations. J. Math. Anal. Appl. 317, 565–582 (2006)
Wang, X., Yang, L., Zhong, C.: Attractors for the nonclassical diffusion equations with fading memory. J. Math. Anal. Appl. 362, 327–337 (2010)
Wang, X., Zhong, C.: Attractors for the non-autonomous nonclassical diffusion equations with fading memory. Nonlinear Anal. 71, 5733–5746 (2009)
Wang, Y., Zhu, Z., Li, P.: Regularity of pullback attractors for non-autonomous nonclassical diffusion equations. J. Math. Anal. Appl. 459, 16–31 (2018)
Xie, Y., Li, Q., Zhu, K.: Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity. Nonlinear Anal., Real World Appl. 31, 23–37 (2016)
Zelik, S.: Asymptotic regularity of solutions of a non-autonomous damped wave equation with a critical growth exponent. Commun. Pure Appl. Anal. 3, 921–934 (2004)
Zhu, K., Xie, Y., Zhou, F.: Attractors for the nonclasssical reaction-diffusion equations on time-dependent spaces. Bound. Value Probl. 2020, 95 (2020)
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The authors would like to thank the reviewers and the editors for their valuable suggestions and comments.
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This work is supported by the National Natural Science Foundation of China (Grant No. 11961059, 11961039, 12161071), Gansu Provincial Department of Education: University Teachers Innovation Fund Project (Grant No. 2023A-046), Science and Technology Programe of Gansu Province (Grant No. 23JRRA908) and the Youth Scientific Research Fund of Lanzhou Jiaotong University (Grant No. 2021022).
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Wang, J., Ma, Q., Zhou, W. et al. Dynamic of the nonclassical diffusion equation with memory. Bound Value Probl 2023, 79 (2023). https://doi.org/10.1186/s13661-023-01767-6
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DOI: https://doi.org/10.1186/s13661-023-01767-6