Abstract
The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.
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1 Introduction
For dissipative dynamical systems generated by partial differential equations or delay differential equations, the phase space is generally not locally compact. One main technique to analyze the complex dynamics of these systems is to find the largest bounded invariant set of the system which is closed and attracts all bounded sets in the phase space, i.e., the global attractor. Moreover, if one can prove that the global attractor has finite dimension, so that, even though the initial phase space is infinite-dimensional, the dynamics, reduced to the global attractor, is, in some proper sense, finite-dimensional and can be described by a finite number of parameters. Therefore, the existence and dimension estimations of global attractors for infinite-dimensional dynamical systems, especially for a large class of parabolic partial differential equations and delay differential equations, have drawn much attention from pure and applied mathematics community during the past decades. See, for instance, the monographs by Babin and Vishik (1992), Hale (1988), Ladyzhenskaya (1991), Robinson (2001) and Temam (1988).
Nevertheless, it is pointed out in Efendiev et al. (2000) and Efendiev et al. (2011) that the global attractor has several drawbacks. First, it may attract the trajectories slowly and, in general, it is very difficult, if not impossible, to express the convergence rate in terms of the physical parameters of the problem. A second shortcoming is that the global attractor may be sensitive to perturbations, which severely limits the application scope since the systems are only approximations of real world models. Moreover, in many situations, global attractors may not be observable in experiments or in numerical simulations because of its complicated geometric structure and may fail to capture important transient behaviors. Hence, Eden, Foias, Nicolaenko and Temam proposed in Eden et al. (1994) the concept of exponential attractor where the theory was established based on the squeezing property in Hilbert spaces which was then extended by Babin and Nicolaenko in Babin and Nicolaenko (1995) to investigate exponential attractors of reaction–diffusion systems in an unbounded domain, and recently generalized by Zhou (2017) and Zhou and Zhao (2016) to investigate the random exponential attractors for non-autonomous stochastic lattice systems with multiplicative white noise and stochastic semilinear wave equation in Hilbert spaces. Generally speaking, an exponential attractor is a compact subset with finite fractal dimension, which is positively invariant and attracts all bounded subsets at an exponential rate. It is well-known that if exponential attractors exists, then they contain global attractors. Although they may be larger than the global attractors, they are more robust than global attractors under perturbations due to the exponential rate of convergence. Hence, they play significant roles in investigating asymptotic behavior of infinite-dimensional nonlinear dynamical systems especially for those with fast convergence rate.
There are many evolution equations arising from real-world models defined in Banach spaces, such as delayed differential equations (Hale and Lunel 1993) and delayed partial differential equations (Wu 1996; Xu et al. 2021, 2022). Therefore, one natural question arise, how to construct exponential attractors for systems in Banach spaces? Efendiev, Miranville and Zelik in Efendiev et al. (2000) developed an alternative method and explicit construction of exponential attractors for semigroups in Banach spaces by using the so-called smoothing property of the semigroup, which originates from Ladyzhenskaya (1985) for proving the finite dimensionality of global attractors. The main ingredient of their works is the following smooth property between two spaces
where Z is a second Banach space which is compactly embedded into X, which has to hold for some \(\tau ^*>0\) and on some bounded positively invariant subset of X. The method was then extended by Czaja, Efendiev, Miranville and Czaja and Efendiev (2011) to construct exponential and uniform attractors for systems in Banach spaces, which has also been widely used to estimate the fractal dimension and construct exponential attractors for deterministic (Carvalho and Sonner 2013; Czaja and Efendiev 2011; Efendiev et al. 2011; Efendiev and Zelik 2008; Hammami et al. 2020; Netchaoui et al. 2021) and random systems (Caraballo and Sonner 2017; Langa et al. 2010; Shirikyan and Zelik 2013; Wang and Zhou 2018).
Although this method is effective for systems in Banach spaces, the construction cannot provide explicit bounds of the fractal dimensions since it depends on the choice of another embedding space which may vary from space to space. Furthermore, the dimension estimation depends on the entropy number between two spaces for which is generally quite difficult, if not impossible, to obtain an explicit bound. Only for some specific examples can the explicit entropy number be obtained, see, for instance (Kolmogorov and Tikhomirov 1993; Triebel 1978; Zelik 2001). Indeed, in the very recent works (Hammami et al. 2020) and (Netchaoui et al. 2021), the authors pointed out only for scalar equations the entropy number of the embedding \(C \hookrightarrow C^1\) is explicitly known, which yields an estimate for the fractal dimension of the exponential attractors. Hence, one naturally wonders whether we can construct exponential attractors with explicit bounds of fractal dimensions for systems in Banach spaces that only depend on the inner characteristics of the system. In our recent work (Hu and Caraballo 2023), we affirm this by extending Eden, Foias, Nicolaenko and Temam’s work (Eden et al. 1994) to autonomous systems in Banach spaces with applications to functional differential equations. In this paper, we go one step further to construct pullback exponential attractors with explicit fractal dimensions of non-autonomous dynamical systems in Banach spaces.
It should be pointed out that in Dung and Nicolaenko (2001), the authors also established exponential attractors in Banach spaces with explicit bounds by the idea originated from Mané (1981). Their results are obtained under the assumption that the semiflow is \(C^1\) and the linearized semiflow at every point inside the absorbing ball can be split into the sum of a compact operator plus a contraction, which were further generalized by Zhong and Zhong (2012) to relax the condition of existence of compact absorbing sets to existence of bounded absorbing sets. Actually, squeezing properties are omnipresent in systems whose linear parts admit exponential dichotomies, which plays significant roles in study invariant manifolds of nonlinear dynamical systems, while the verification of \(C^1\) smoothness may be tedious, especially for functional differential equations. Therefore, in this paper, we establish a new method by combining the squeezing property proposed in early work of Foias and Temam (1979) and the covering lemma of the finite-dimensional subspace of Banach space established in Mané (1981). We do not need the strict restriction of the boundedness of derivative of the semiflow and we also obtain explicit bounds on the dimension of the invariant set in Banach spaces. The main contributions of this work are in the following three folds.
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Unlike the works (Mallet-Paret 1976; Qin and Su 2019; So and Wu 1991), where the functional differential equations are recast into Hilbert spaces to study the dimensions of global attractors, we directly propose a construction procedure of pullback exponential attractors in the natural spaces of functional differential equations, i.e., Banach spaces.
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Compared with the recent works (Hammami et al. 2020) and (Netchaoui et al. 2021), where Banach spaces are taken as the phase spaces, we derive explicit bounds on fractal dimensions of the constructed pullback exponential attractors that only depend on the spectrum of the linear parts and the Lipschitz constants of the nonlinear parts while not related to the entropy number.
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Different from the early works (Dung and Nicolaenko 2001; Hu and Caraballo 2023; Zhong and Zhong 2012), where autonomous systems are studied, we investigated non-autonomous systems generated by different kinds of non-autonomous partial functional differential equations in this work. Specifically, we will consider both non-autonomous linear part and non-autonomous nonlinear part by semigroup approach or variational technique. Even more, the approach used here is quite different from Dung and Nicolaenko (2001); Zhong and Zhong (2012).
The outline of our paper is as follows. In Sect. 2, we recall basic notions and results from the theory of infinite-dimensional dynamical systems, introduce the notion of pullback exponential attractors and propose the construction procedure of pullback exponential attractors. In Sect. 3, we construct pullback exponential attractors for non-autonomous retarded reaction–diffusion equations. We consider two situations, i.e., the non-autonomous effect appears in the linear part and the nonlinear part respectively. For the former scenario, we prove the squeezing property under the assumption that the process generated by the linear part admits an exponential dichotomy, while in the latter situation, we adopt a variational technique. Indeed, even the existence of pullback attractors is new for the non-autonomous retarded reaction–diffusion equations. Sections 4 and 5 are devoted to applications of the theoretical results to the retarded 2D Navier–Stokes equation and the retarded semilinear wave equation by variational techniques, since for these two equations, it is quite difficult to show the linear parts generate semigroups in the natural phase space. At last, we summarize the paper and point out some potential directions for future research in Sect. 6.
2 Pullback Exponential Attractor
We first introduce some preliminaries for establishing our main results, including the definitions of evolution process, pullback exponential attractors as well as some hypotheses.
Definition 2.1
A two-parameter set of mappings \(\{U(t, s): t, s \in \mathbb {R}, t \geqslant s\}\) acting on X, i.e., \(U(t, s): X \rightarrow X\) for all real numbers \(t, s \in \mathbb {R}\) with \(t \geqslant s\), is said to be an evolution process on X if it satisfies
where \(\textrm{Id}_X: X \rightarrow X\) represents the identity map on X. Moreover, if
is continuous, where \(\mathcal {T}:=\{(t, s) \in \mathbb {R} \times \mathbb {R} \mid t \ge s\}\), then it is called a continuous evolution process.
We now give the following definition of pullback attractor and pullback exponential attractors.
Definition 2.2
Let \(\{U(t, s): t, s \in \mathbb {R}, t \geqslant s\}\) be an evolution process in X. A family of non-autonomous sets \(\mathcal {A}=\{\mathcal {A}(t) \mid t \in \mathbb {R}\}\) is said to be the global pullback attractor for U if it satisfies the following properties:
-
(i)
\(\mathcal {A}(t) \subset X\) is non-empty and compact for all \(t \in \mathbb {R}\),
-
(ii)
\(\mathcal {A}\) is strictly invariant, i.e.,
$$\begin{aligned} U(t, s) \mathcal {A}(s)=\mathcal {A}(t) \quad \forall t \ge s, s \in \mathbb {R}, \end{aligned}$$ -
(iii)
\(\mathcal {A}\) pullback attracts all bounded sets, i.e., for every bounded subset \(D \subset X\) and \(t \in \mathbb {R}\),
$$\begin{aligned} \lim _{s \rightarrow \infty } {\text {dist}}_H(U(t, t-s) D, \mathcal {A}(t))=0, \end{aligned}$$and \(\mathcal {A}\) is minimal within the families of closed subsets that pullback attract all bounded subsets of X.
Definition 2.3
Let \(\{U(t, s): t, s \in \mathbb {R}, t \geqslant s\}\) be an evolution process in X. The family of non-empty compact subsets \(\mathcal {M}=\{\mathcal {M}(t) \mid t \in \mathbb {R}\}\) is called a pullback exponential attractor for the evolution process \(\{U(t, s): t, s \in \mathbb {R}, t \geqslant s\}\) if
-
(i)
\(\mathcal {M}\) is positively invariant, i.e.,
$$\begin{aligned} U(t, s) \mathcal {M}(s) \subset \mathcal {M}(t) \quad \forall t \ge s. \end{aligned}$$ -
(ii)
the fractal dimension of the sections \(\mathcal {M}(t), t \in \mathbb {R}\), is uniformly bounded, i.e.,
$$\begin{aligned} \sup _{t \in \mathbb {R}}\left\{ {\text {dim}}_f(\mathcal {M}(t))\right\} <\infty , \end{aligned}$$where \({\text {dim}}_f(\mathcal {M}(t))<\infty \) is defined as
$$\begin{aligned} {\text {dim}}_f(\mathcal {M}(t))=\lim _{\varepsilon \rightarrow 0} \frac{\ln \left( N_{\varepsilon }^X(\mathcal {M}(t))\right) }{\ln \left( \frac{1}{\varepsilon }\right) }, \end{aligned}$$and \(N_{\varepsilon }^X(\mathcal {M}(t))\) denotes the minimal number of \(\varepsilon \)-balls in X with centers in \(\mathcal {M}(t)\) needed to cover A.
-
(iii)
\(\mathcal {M}\) exponentially pullback attracts all bounded sets, i.e., there exists a constant \(\omega >0\) such that for every bounded subset \(D \subset X\) and every \(t \in \mathbb {R}\)
$$\begin{aligned} \lim _{s \rightarrow \infty } e^{\omega s} {\text {dist}}_X(U(t, t-s) D, \mathcal {M}(t))=0, \end{aligned}$$where \(\textrm{dist}_X(A,B)\) denotes the Hausdorff semi-distance defined by between A and B, defined as
$$\begin{aligned} \textrm{dist}_X(A, B)=\sup _{a \in A} \inf _{b \in B} d(a, b), \quad \text{ for } A, B \subseteq X. \end{aligned}$$
Definition 2.4
A family of time dependent bounded subsets \(\mathcal {B}(t),t \in \mathbb {R}\) is said to grow at most sub-exponentially in the past provided
where \(R_{\mathcal {B}(t)}\) denotes the diameter of \(\mathcal {B}(t)\subset X\).
As in the autonomous case, the existence of compact absorbing sets is the crucial property in order to obtain pullback attractors. For the following result, see (Crauel and Flandoli 1994).
Lemma 2.1
Let \(\{U(t, s): t, s \in \mathbb {R}, t \geqslant s\}\) be a two-parameter process, and suppose \(U(t, s): X \rightarrow X\) is continuous for all \(t \geqslant s\). If there exists a family of compact (pullback) absorbing sets \(\{B(t)\}_{t \in \mathbb {R}}\), then there exists a pullback attractor \(\{\mathcal {A}(t)\}_{t \in \mathbb {R}}\), and \(\mathcal {A}(t) \subset B(t)\) for all \(t \in \mathbb {R}\). Furthermore,
where
For a finite-dimensional subspace F of a Banach space X, denote by \(B^F_r(x)\) the ball in F of center x and radius r, that is \(B^F_r(x)=\{y \in F |\Vert y-x\Vert \le r\}\). For later use, we introduce the following covering lemma of balls in finite-dimensional Banach spaces which was proved in Mané (1981).
Lemma 2.2
For every finite-dimensional subspace F of a Banach space X, we have
for all \(r_1>0, r_2>0\), where \(m={\text {dim}} F\) and \(N\left( r_1, B_{r_2}^F(0)\right) \) is the minimum number of balls needed to cover the ball of radius \(r_1\) by balls \(B_{r_2}^F(0)\) of radius \(r_2\) calculated in the metric space X.
For notation simplicity, we will write \(\{U(t, s): t, s \in \mathbb {R}, t \geqslant s\}\) simply as \(\{U(t, s)\}\) in the following. In order to construct the pullback exponential attractor we need to impose the following assumptions on the process \(\{U(t, s)\}\).
\(\left( \mathcal {H}_1\right) \) For the process \(\{U(t, s)\}\), there exists a family of bounded sets \(\mathcal {B}(t) \subset X\), \(t \in \mathbb {R}\), that pullback absorbs all bounded subsets of X. That is, for all bounded subsets \(D \subset X\) and all \(t \in \mathbb {R}\) there exists \(T_{D, t}>0\) such that
\(\left( \mathcal {H}_2\right) \) The family of bounded sets \(\mathcal {B}(t) \subset X, t \in \mathbb {R}\) is positively invariant, that is, \(U(t, s)\mathcal {B}(s) \subseteq \mathcal {B}(t)\) for all \(t \ge s\).
\(\left( \mathcal {H}_3\right) \) There is a finite-dimensional projection \(P(t):X\rightarrow P(t)X\) with finite dimension
and there are three positive numbers \(M_1, M_2, M_3\) and two constants \(\lambda _0\) and \(\lambda _1\) such that
and
for any \(t\in \mathbb {R}\) and some \(s_0\ge 0\) and \(\varphi , \psi \) in \(\mathcal {B}(t)\).
In the following, we are devoted to the construction of exponential attractors for the discrete evolution process \(\{U(n,m) \}\).
Theorem 2.1
Let \(\{U(n,m) \}\) be a discrete evolution process in X and the assumptions \(\left( \mathcal {H}_1\right) -\left( \mathcal {H}_3\right) \) are satisfied for discrete times \(n, m \in \mathbb {Z}\). Moreover, we assume that the diameter of the family of absorbing sets \(\{\mathcal {B}(n)\}, n \in \mathbb {Z}\), grows at most sub-exponentially in the past, and there exists \(0<\alpha <M_1\) such that \(\zeta :=\alpha e^{\lambda _0}+M_2e^{\lambda _1}+M_3e^{\lambda _0}<1\). Then, there exists a pullback exponential attractor \(\{\mathcal {M}(n)\}\) for the semigroup \(\{U(n,m) \}\), and the fractal dimension is bounded by
Proof
1) Covering of \(U(n, n-m)\mathcal {B}(n-m)\). We construct the covering of \(U(n, n-m)\mathcal {B}(n-m)\) by inductively defining a family of sets \(W^m(n)\) in \(m\in \mathbb {N}^+\) that depend on the time instant \(n\in \mathbb {Z}\) and satisfy the following properties
where \(\sharp W^m(n) \) represents the number of elements of \(W^m(n)\).
We first consider the case \(m=1\), i.e., we construct a covering of the image when \(U(n, n-1)\mathcal {B}(n-1)\). Denote by \(R_{\mathcal {B}(i)}:=\sup _{u \in \mathcal {B}(i)}\Vert u\Vert _X, i=n-m, n-(m-1),\cdots , n-1\), then for any \(u_{1}\in \mathcal {B}(n-1)\), we have \(\mathcal {B}(n-1)\subset B_{R_{\mathcal {B}(n-1)}}\left( u_{1}\right) \). For any \(u \in \mathcal {B}(n-1)\cap B\left( u_{1}, R_{\mathcal {B}(n-1)}\right) \), it follows from \(\left( \mathcal {H}_3\right) \) that
and
By Lemma 2.2, we can find \(y_{1}^1, \ldots , y_{1}^{n_1}\) such that
with
where \(\Lambda \) is the dimension of P(n)X and we have denoted by \(B_{P(n)X}(y, r)\) the ball in P(n)X of radius r and center y. Set
for \( j=1, \ldots , n_1\) and \(W^1=\{u_{1}^1, u_{1}^2,\cdots , u_{1}^{n_1}\}\). Then, for any \(u \in \mathcal {B}(n-1)\cap B\left( u_{1}, R_{\mathcal {B}(n-1)}\right) \), there exists a j such that
indicating (W3) is satisfied for \(m=1\). Furthermore, it is clear from the definition of \(W^1\) that it satisfies (W1) and (W2). This completes the proof of the case \(m=1\).
Assume that the sets \(W^l(n)\) satisfying (2.7) have already been constructed for all \(m\le l\) and \(n \in \mathbb {Z}\), i.e., there exists covering
We construct in the sequel the covering of \(W^{l+1}(n)\) satisfying (2.7). By the process property and the induction hypothesis (W3), we have
In other words, \(U(n, n-(l+1)) \mathcal {B}(n-(l+1))\) can be covered by \(\bigcup _{u \in W^l(n-1)} U(n, n-1) B_{\zeta ^lR_{\mathcal {B}(n-l-1)}}(u)\). We construct in the following a covering of \(\bigcup _{u \in W^l(n-1)} U(n, n-1) B_{\zeta ^lR_{\mathcal {B}(n-l-1)}}(u)\).
Let \(u_l \in W^l(n-1)\). It follows from induction hypothesis (W1) that \(u_l \in W^l(n-1) \subset U(n-1, n-1-l) \mathcal {B}(n-l-1)\subset \mathcal {B}(n-1)\). Therefore, for any \(u \in \mathcal {B}(n-1)\cap B\left( u_l, \zeta ^lR_{\mathcal {B}(n-l-1)}\right) \), it follows from \(\left( \mathcal {H}_3\right) \) that
and
By Lemma 2.2, we can find \(y_l^1, \ldots , y_l^{n_l}\) such that
with
where \(\Lambda \) is the dimension of P(n)X and we have denoted by \(B_{P(n)X}(y, r)\) the ball in P(n)X of radius r and center y. Set
for \( j=1, \ldots , n_l\). Then, it follows from (2.16)–(2.18) that, for any \(u \in \mathcal {B}(n-1)\cap B\left( u_l, \zeta ^lR_{\mathcal {B}(n-l-1)}\right) \), there exists a j such that
This implies that \(\bigcup _{u \in W^l(n-1)} U(n, n-1) B_{\zeta ^lR_{\mathcal {B}(n-l-1)}}(u)\) is covered by balls with radius \(\zeta ^{l+1}R_{\mathcal {B}(n-l-1)}\) and centers \(\{u_l^1, u_l^2, \cdots , u_l^{n_l}\}\) and hence (W3) holds. Denote the new set of centers by \(W^{l+1}(n)\). From the induction hypothesis, we have \(\sharp W^{l}(n)\le [\Lambda 2^\Lambda \left( 1+\frac{M_1}{\alpha }\right) ^\Lambda ]^l\), which yields \(\sharp W^{l+1}(n) \le n_l \sharp W^{l}(n) \le [\Lambda 2^\Lambda \left( 1+\frac{M_1}{\alpha }\right) ^\Lambda ]^{l+1}\) and proves (W2). By construction the set of centers \(W^{l+1}\)(n), we can see \(W^{l+1}(n) \subset U(n, n-(l+1)) \mathcal {B}(n-(l+1))\), which concludes the proof of the properties (W1).
2) Construction of random exponential attractor for \(\{U(n,m) \}\). We define \(E^1(n):=W^1(n)\) and set
Then, if follows from the definition of the sets \(E^m(n)\), the properties of the sets \(W^m(n)\) and the positive invariance of the absorbing set \(\mathcal {B}(n)\) that the family of sets \(E^m(n), m \in \mathbb {N}^+\) satisfies
-
(E1)
\(U(n,n-1) E^m(n-1) \subset E^m(n), \quad E^m(n) \subset U(n,m)\mathcal {B}(m)\),
-
(E2)
\(E^m(n)=\bigcup _{i=0}^n U(n, n-i) W^{m-i}(n-i), \quad \sharp E^m(n) \le \sum _{i=0}^n (\Lambda 2^\Lambda \left( 1+\frac{ M_1}{\alpha }\right) ^\Lambda )^i\),
-
(E3)
\(U(n,m) \mathcal {B}(m)\subset \bigcup _{u \in E^m(n)} B_{\zeta ^n R_{\mathcal {B}}}(u)\).
Based on the family of sets \(E^m(n)\), we define \(\mathcal {M}(n):=\overline{\bigcup _{m \in \mathbb {N}^+} E^m(n)}\) and show that it yields an exponential attractor for the semigroup \(\{U(n,m)\}\).
Positive invariance of \(\mathcal {M}(n)\). It follows from property (E1) that, for all \(l \in \mathbb {Z}\), we have
Thanks to the continuity property in \(\left( \mathcal {H}_1\right) \), we can take closure in both sides of (2.23), giving rise to
Compactness and finite dimensionality of \(\mathcal {M}(n)\). We prove in the sequel that the set \(\mathcal {M}(n)\) is non-empty, precompact and of finite fractal dimension. It follows from (E1) that, for any \(l \in \mathbb {Z}\) and \(m\ge l+1\),
Thus, for any \(l \in \mathbb {Z}\), we have
Since \(\zeta <1\), for any given \(\varepsilon >0\), there exists \(l \in \mathbb {Z}\) such that
which combined with the fact
indicates that the estimate of the number of \(\varepsilon \)-balls in X needed to cover \(\bigcup _{m \in \mathbb {N}^+} E^m(n)\) is
This proves the precompactness of \(\bigcup _{m \in \mathbb {N}^+} E^m(n)\) in X, which directly implies the closure \(\mathcal {M}(n):=\overline{\bigcup _{m \in \mathbb {N}^+} E^m(n)}\) is compact in X since X is a Banach space.
It follows from (2.27) and (2.29) that the fractal dimension of the set \(\mathcal {M}(n)\) can be estimated by
for any \(\varsigma >0\), where the last inequality follows from the assumption that \(R_{\mathcal {B}(n-l+1)}\) grows at most sub-exponentially in the past. Due to the arbitrariness of \(\varsigma >0\), we have
3) Exponential attraction of \(\mathcal {M}(n)\). It remains to show that the set \(\mathcal {M}(n)\) exponentially attracts all bounded subsets of X at time \(n \in \mathbb {Z}\). It follows from assumptions \(\left( \mathcal {H}_1\right) \) that, for any bounded subset \(D \subset X\), there exists an \(n_{D,n} \in \mathbb {Z}\) such that \(U(k,k-l) D \subset \mathcal {B}(k)\) for all \(l \ge n_{D,n}\) and \(k\le n\). If \(l \ge n_{D,n}+1\), that is \(l=n_{D,n}+n_0\) with some \(n_0 \in \mathbb {N}\), then
for some constants \(c \ge 0\) and \(\omega >0\), since \(\zeta <1\). This completes the proof. \(\square \)
By adopting the same procedure as the proof of Theorem 3.2 in Carvalho and Sonner (2013), we have the following results about the existence of exponential attractors for evolution process in Banach spaces.
Theorem 2.2
For the evolution process \(\{U(t,s)\}\) on the Banach space X and assume that hypothesis \(\left( \mathcal {H}_1\right) \)–\(\left( \mathcal {H}_3\right) \) hold and conditions of Theorem 2.1 hold. Then, there exists a pullback exponential attractor \(\mathcal {M}(t)\) for the evolution process \(\{U(t,s)\}\), and the fractal dimension is bounded by
Remark 2.1
By (2.33), we can see the fractal dimension of the pullback exponential attractors constructed in theorem 2.2 depends on the parameter \(\alpha \). Generally, if we take \(\alpha \uparrow M_1\) and assume \(M_1e^{\lambda _0}+M_2e^{\lambda _1}+M_3e^{\lambda _0}<1\), then for all \(\alpha \in (0,M_1)\), we have \(\alpha e^{\lambda _0t_0}+ M_2e^{\lambda _1t_0}+ M_3e^{\lambda _0t_0}<1\) and hence we get an \(\alpha \)-independent estimation
3 Retarded Reaction–Diffusion Equation
This section is devoted to the existence of pullback exponential attractors for non-autonomous retarded reaction–diffusion equations with asymptotically autonomous linear part or non-autonomous nonlinear part.
3.1 Asymptotic Autonomous Linear Part
We first consider the following non-autonomous retarded reaction–diffusion equation on bounded domain with Dirichlet boundary condition and asymptotically autonomous linear part
where \(a, c, l \in C_b^\mu \left( \mathbb {R}_{+}, C[0, \pi ]\right) , \mu \in (0,1)\), the space of uniformly bounded, \(\mu \)-Hölder continuous \(C[0, \pi ]\)-valued functions, \(a(t, x) \ge a_0>0\) and
uniformly for \(x \in [0, \pi ]\) as \(t \rightarrow \infty \). Let \(H=L^2(0, \pi )\) with inner product \((\xi ,\eta )=\int _0^\pi \xi (x)\eta (x)dx\), norm \(\Vert \xi \Vert _H=[\int _0^\pi \xi ^2(x)dx]^{1/2}\) for any \(\xi , \eta \in H\) and \(X=C([-r, 0],H)\) the continuous function from \([-r, 0]\) to H endowed with the supremum norm \(\Vert \phi \Vert _{X}=\sup _{\theta \in [-r, 0]}\Vert \phi (\theta )\Vert _H\) for any \(\phi \in X\). Let \(D=\) \(W^{2,2}(0, \pi ) \cap W_0^{1,2}(0, \pi )\) be endowed with the usual norm. Set
for \(\varphi \in D\) and \(t \ge 0\). On X, we further define
and define \(A_U: X\rightarrow X\) by
for any \(\phi \in X\). It follows from Wu (1996) that the characteristic values of the linear part \(A_U\) are the roots of the following characteristic equation
Since \(A_U\) is compact, it follows from Wu (1996) [Theorem 1.2 (i)] that the spectrum of \(A_U\) is point spectrum, which we denote by \(\varrho _1>\varrho _2>\cdots \) with multiplicity \(n_1, n_2,\cdots \), where \(\varrho _1\) is defined as
In the following, we always assume that \(l-c<1\) and it follows from Wu (1996)[Lemma 1.13, p. 73] that if \(c>0\), \(l>0\) and \(l-c<1\), then \(\varrho _1<0\). By the results in Schnaubelt (2004)[p. 3541], the evolutions process \(S(t,\sigma ):\mathbb {R}\times \mathbb {R}\times X\rightarrow X\) defined by \(S(t,\sigma )\phi =u_t^\phi (\cdot ,\sigma )\) with \(u^\phi (t,\sigma )\) being the solution of the following linear part of (3.1)
which admits an exponential dichotomy with a two-dimensional stable subspace, that is, there exist a positive constants K and a negative constant \(\beta <0\), and a m dimension projection operators \(P(t): X \rightarrow X_m, s \in \mathbb {R}\) and \(Q(t)=I-P(t):X \rightarrow X_m^\bot , t \in \mathbb {R}\) such that
Moreover, by definition of \(\varrho _1\), there exist positive constants \(-\gamma <\varrho _1\) and \(K_{0}\) such that
for all \(t \ge \sigma \). We assume that f satisfies the following global Lipschitz condition.
\(\mathbf {Hypothesis\ A3}\) \(\left\| f\left( \phi _1\right) -f\left( \phi _2\right) \right\| \le L\left\| \phi _1-\phi _2\right\| \text{ for } \text{ any } \phi _1, \phi _2 \in X.\)
It follows from Theorem 2.6 in Wu (1996) and some standard contraction techniques, one can see under assumption \(\mathbf {Hypothesis \ A3}\), the non-autonomous nonlinear equation (3.1) admits a solution \(u^\phi (t,\sigma )\) for any \(t\in [\sigma -r, \infty )\), which is also continuous with respect to the initial condition and can be represented as
where \(X_0:[-r, 0] \rightarrow B(X, X)\) is given by \(X_0(\theta )=0\) if \(-r \le \theta <0\) and \(X_0(0)=I d\), where B(X, X) is the family of bounded linear operators on X.
Define the non-autonomous evolution process generated by (3.1) by \(\Phi (t,\sigma )\phi =u^\phi _t(\cdot , \sigma )\) for any \(\phi \in X\), which is continuous for any \(t\ge \sigma \). In the following, we construct exponential attractors for \(\Phi (t,\sigma )\). By similar techniques as those in the proof of Theorem 3.1 from Hu and Caraballo (2023b), we can see that \(\Phi (t,\sigma )\) admits a family of positively invariant absorbing sets \(\mathcal {B}(\sigma )\) for any \(\sigma \in \mathbb {R}\), implying \(\left( \mathcal {H}_1\right) \) and \(\left( \mathcal {H}_2\right) \) hold.
Theorem 3.1
Assume that \(\mathbf {Hypothesis \ A4}\) as well as \(\mathbf {Hypothesis \ A3}\) hold, \(K_0<1\) and \(K_0L_f-\gamma <0\). Then, the dynamical system \(\Phi \) admits an invariant absorbing set \(\mathcal {B}(\sigma )\) defined by
Subsequently, we prove the squeezing property of \(\Phi \), i.e., \(\left( \mathcal {H}_3\right) \) holds.
Theorem 3.2
Let P be the finite-dimensional projection defined by (3.6), \(K, \beta ,\gamma \) and \(K_0\) being defined by (3.6) and (3.5) respectively and assumptions of Theorem 3.1 hold. Then, we have
and
for any \(t\ge 0\) and \(\varphi , \psi \in \mathcal {B}\).
Proof
For any \(\varphi , \psi \in X\), denote by \(y=\varphi -\psi \) and \(w_t(\cdot ,\sigma )=\Phi (t,\sigma )\varphi -\Phi (t,\sigma )\psi =u^\varphi _t(\cdot , \sigma )-u^\psi _t(\cdot , \sigma )\). Then, it follows from (3.10) that
Taking projection \(I-P(t)\) on both sides of (3.12) leads to
Multiplying both sides of (3.13) by \(e^{\gamma (t-\sigma )}\) implies
Applying the Gronwall inequality, we have
indicating that
Hence, the second part holds with \(\lambda _0=L_fK_0-\gamma \), \(\lambda _1=\beta \), \(M_2=K\) and \(M_3=\frac{KL_fK_0 }{-\beta -\gamma +L_fK_0}\).
Subsequently, we prove the first part. Since \(S(t,\sigma )y=P(t)S(t,\sigma )y+(I-P(t))S(t,\sigma )y\), we have
Taking projection P(t) on both sides of (3.12) and on account of (3.17) yields
Multiplying both sides of (3.18) by \(e^{\gamma (t-\sigma )}\) implies
Applying again the Gronwall inequality,
indicating that
Hence, the first part holds by taking \(M_1=(K_0+K)\) and \(\lambda _0=L_fK_0-\gamma \). \(\square \)
By Theorem 2.2, we have the following results about existence of a pullback exponential attractor \(\mathcal {M}(t)\) for the nonlinear dynamical system \(\Phi \) generated by (3.1).
Theorem 3.3
Let P be the finite-dimensional projection defined by (3.6), \(K, \beta ,\gamma \) and \(K_0\) being defined by (3.6) and (3.5), respectively, and assumptions of Theorem 3.1 hold. Moreover, assume there exists \(\alpha >0\) such that \(\zeta :=\alpha e^{(L_fK_0-\gamma )}+Ke^{\beta }+\frac{KL_fK_0 }{-\gamma +L_fK_0-\beta }e^{(L_fK_0-\gamma )}<1\). Then, (3.1) admits an exponential attractor \(\mathcal {M}(t)\) whose fractal dimension satisfies
We have the following special case about the fractal dimension of global attractor \(\mathcal {M}(t)\) in the case \(m=1\).
Corollary 3.1
Let P be the finite-dimensional projection defined by (3.6), \(K, \beta ,\gamma \) and \(K_0\) being defined by (3.6) and (3.5), respectively, and assumptions of Theorem 3.1 hold with \(m=1\). Moreover, assume there exists \(\alpha >0\) such that \(\zeta :=(\alpha +K) e^{(L_fK_0-\gamma )}+Ke^{\varrho _1}<1\). Then, (3.1) admits an exponential attractor \(\mathcal {M}(t)\) whose fractal dimension is bounded as
3.2 Non-autonomous Nonlinear Term
Subsequently, we consider the situation in which the non-autonomous effect comes from the nonlinear term, i.e., the following non-autonomous initial boundary value problem with delay
Here, a, b are positive constants, f is the nonlinear delayed forcing term and h(x) is the time independent and spatial dependent external force. The uniform attractors of (3.24) on unbounded domain when the non-autonomous term does not depend on \(u_t\) has been investigate in Wang and Kloeden (2014). Here, we restrict ourselves to the domain \([0,\pi ]\) and pay attention to the existence as well as the topological dimensions estimation of pullback attractors.
Due to the non-autonomous effect accurses in the nonlinear term of (3.24), we do not use the semigroup approach as Sect. 3.1 but adopt the variational one developed in Caraballo and Real (2004). We first introduce more notations that will be used in the remaining of this section. Let \(C_0^{\infty }(0,\pi )\) be the space of smooth function on \((0,\pi )\) with compact support. Set \(\mathcal {V}=\left\{ u \in C_0^{\infty }(0,\pi ): {\text {div}} u=0\right\} \) and denote by H the closure of \(\mathcal {V}\) in \(L^2(0,\pi )\) with inner product \((\cdot , \cdot )\) defined by \((u, v)=\int _0^\pi u(x) v(x) \textrm{d} x\) and norm \(|\cdot |\) defined by \(|u|=(u,u)^{1/2}\) for any \(u, v \in L^2(0,\pi )\). Let V be the closure of \(\mathcal {V}\) in \(H_0^1(0,\pi )\) with scalar product \(((\cdot , \cdot ))\) defined by \(((u, v))=\int _0^\pi \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} \mathrm {~d} x\) and norm \(\Vert \cdot \Vert \) defined by \(\Vert u\Vert =((u,u))^{1/2}\) for any \(u, v \in H_0^1(0,\pi )\). Denote by \(L_H=L^2([-r, 0],H)\). By the standard theory of Soblev spaces, one can see \(V \subset H \equiv H^{\prime } \subset V^{\prime }\), where \(H^{\prime } \) and \(V^{\prime }\) are the dual spaces of V and H respectively and the injections are dense and compact. Denote by \(\langle \cdot , \cdot \rangle \) the duality pairing between V and \(V^{\prime }\) and \(X=C([-r,0],H)\) with the usual supremum norm \(\Vert \cdot \Vert _X\). Set \(D(A)=H^2 \cap V\), then \(A u={\tilde{P}} \Delta u, \forall u \in D(A)\), (\({\tilde{P}}\) the ortho-projector from \(L^2(0,\pi )\) onto H.
Let \(0<\mu _1<\mu _2<\cdots<\mu _m<\cdots \) be the eigenvalues of \(-A\) with eigenfunctions \(e_1,e_2,\cdots ,e_m,\cdots \). Denote by \(V_m=\textrm{span} \{e_1, e_2, \cdots , e_m\}\) and \(V_m^\bot =\textrm{span} \{e_{m+1}, e_{m+2}, \cdots \}\) the finite-dimensional space spanned by \(\{e_1, e_2, \cdots , e_m\}\) and its orthogonal complement, respectively. Let \(P_m:V\rightarrow V_m\) be the finite projection of V onto \(V_m\) defined by
for any \(\phi \in V\). Then, we have \(\mu _{m+1}|(I-P_m)v|\le \Vert v\Vert =(-Av,v)\) and \(\mu _{1}|v|\le \Vert v\Vert =(-Av,v)\) for any \(v\in H\), indicating that
and
for any \(v\in V\). Let \(C_V=C^0([-r, 0]; V)\) be the space of continuous function from \([-r,0]\) to V equipped with the usual supremum norm. Similar to De Monvel et al. (1998); Chueshov and Scheutzow (2001), we define the following m-dimensional projector \(P={\hat{P}}_m\) in \(C_V\) by
from \(C_V\) onto \(C_{V_m}\), based on the projection operator \(P_m\). It follows from (3.28) that
and
Remark 3.1
By De Monvel et al. (1998); Chueshov and Scheutzow (2001), P is the spectral projector of the infinitesimal generator \(\mathscr {A}\) of the linear semigroup \(T_t\) in \(C_V\) defined by
with u(t) being the solution of (3.24) with \(a=b=0\), \(f(t, u(x, t-r))\equiv 0\).
For notation simplicity, denote by \(g(t,\psi )=-b\psi (-r)+f(t, \psi (-r))\) for \(\psi \in X\) and thus, (3.24) can be rewritten in the following abstract form on H
Now, we can analyze problem (3.31) for a more general functional g by imposing the following assumptions, similarly as it was done in Caraballo and Real (2004). Let us consider \(g: \mathbb {R} \times X \rightarrow H\) satisfying: \(\mathbf {Hypothesis\ A4}\) For all \(\xi \in X\), \(g(\cdot , \xi ):\mathbb {R} \rightarrow H\) is measurable.
\(\mathbf {Hypothesis\ A5}\) For all \(t \in \mathbb {R}\), \(g(t, 0)=0\),
\(\mathbf {Hypothesis\ A6}\) There exists \(L_g>0\) such that \(\forall t \in \mathbb {R}, \forall \xi , \eta \in X\)
\(\mathbf {Hypothesis\ A7}\) There exists \(m_0 \geqslant 0, C_g>0\) such that for all \(l \in \left[ 0, m_0\right] , \tau \leqslant t, u\) and \(v \in C([\tau -r, t]; H)\), the continuous function space from \([\tau -r, t]\) to H, the following inequality holds
Following similar techniques as Douady and Oesterl (1980)[Theorem 2.3] and Wang and Kloeden (2014)[Theorem 8], we have the following results on the existence of solutions.
Lemma 3.1
Assume that \(\mathbf {Hypothesis\ A4-A7}\) hold. Then, for each \(\tau \) \(\in \mathbb {R}\)
-
(i)
for any \(\phi \in X\), there exists a solution \(u(\cdot )\) to problem (3.24) with \(u \in L^2(\tau -r, T; H) \cap L^2(\tau , T; V) \cap L^{\infty }(\tau , T; H) \cap C([\tau -r, T]; H), \forall T>\tau \),
-
(ii)
for any \(\phi \in C([\tau -r, T]; V)\), problem (3.24) admits a strong solution
$$\begin{aligned} u \in L^2\left( \tau , T; H\right) \cap C([\tau -r, T]; V), \quad \forall T>\tau . \end{aligned}$$
It follows from Lemma 3.1 and Caraballo and Real (2004)[Theorem 9] that the family of mappings \(U(t, \tau ): X \rightarrow X\) defined by
is a continuous process for any \(\phi \in X\) and any \(\tau \leqslant t\). For later analysis, we also introduce the product space \(M_H=H \times L_H\), which is a Hilbert space with associated norm
Then, we can define the corresponding process on \(M_H\) as
Following Remark 7 in Caraballo and Real (2004), we also consider the family of mappings \({\tilde{U}}(\cdot , \cdot ): M_H \rightarrow L_H\) defined as
Observe that
In this way, we then have that the process \(S(t, \tau )\) can be rewritten as
Moreover, define the linear mapping j by
This map is obviously continuous from X into \(H \times X\) and into \(M_H\). Noticing that for all \(\left( u_0, \phi \right) \in M_H\) it holds that \({\tilde{U}}(t, \tau )\left( u_0, \phi \right) \in X\) provided that \(t \geqslant \tau +r\), we then can write
It follows from Theorem 9 in Caraballo and Real (2004) that the above defined \( U (t, \tau ): X\rightarrow X\) and \(S(t, \tau ):M_H\rightarrow M_H\) are both continuous processes for \(t\ge \tau \).
Using similar arguments in Caraballo and Real (2004), we have the following results concerning existence of global pullback attractors of (3.1), implying \(\left( \mathcal {H}_1\right) \) and \(\left( \mathcal {H}_2\right) \) hold. For readers’ convenience, we provide an outline of the proof here. Details can be found in Caraballo and Real (2004).
Theorem 3.4
Assume that \(\mathbf {Hypothesis\ A4-A7}\) hold for any \(\tau \leqslant t\) with \(m_0>0\) and \(l+1+2C_g-2(\mu _1+a)<0\). Then, there exists a unique uniformly bounded pullback attractor \(\{\mathcal {A}(t)\}_{t\in \mathbb {R}}\) for the process \(U(t, \tau )\).
Proof
We first show the existence of a family of bounded absorbing sets \(\{\mathcal {B}(t)\}_{t \in \mathbb {R}}\) of \({\tilde{U}}(t, \tau )\) in X. By Caraballo and Real (2004)[Definition 10], it suffices to prove that for any bounded \({\tilde{D}} \subset M_H\) and \(t\in \mathbb {R}\), there exists \(T_{{\tilde{D}}}(t)\) such that for all \(s\geqslant T_{{\tilde{D}}}(t)\) it holds \({\tilde{U}}(t, t-s){\tilde{D}}(t)\subset B(t)\). Let \({\tilde{D}}\) be bounded in \(M_H\), that is, there exists \(d>0\) such that
Take \(\left( u_0, \phi \right) \in {\tilde{D}}\), \(\tau \in \mathbb {R}\) and denote as usual \(u(\cdot )=u\left( \cdot ; \tau ,\left( u_0, \phi \right) \right) \). Taking inner product on both sides of (3.31) and on account of (3.27), we have
implying that
Choose \(l \in \left( 0, m_0\right) \) such that \(l+2C_g-2(\mu _1+a)<0\). Then,
Integrating from \(\tau \) to \(t (\ge \tau )\),
Thus,
Taking \(t \geqslant \tau +r\), we have for \(\theta \in [-r, 0]\)
Setting time \(t-s\) instead of \(\tau \) and denoting \(u(\cdot )=u\left( \cdot , t-s,\cdot \right) \) lead to
Therefore, there exists sufficient large \(T_{{\tilde{D}}}(t)\) such that for all \(s\geqslant T_{{\tilde{D}}}(t)\) it holds
implying that the balls \(\mathcal {B}(t)=B_{X}\left( 0, 2\frac{\left| h\right| ^2}{l}\right) \) form an absorbing family of bounded sets for the mappings \({\tilde{U}}(t, \tau )\). By Caraballo and Real (2004)[Lemma11], we can see \(\{\mathcal {B}(t)\}_{t \in \mathbb {R}}\) is a family of absorbing sets for \(U(t, \tau )\) in X.
By taking the inner product of (3.31) with \(\Delta u\) and using similar techniques as those in the proofs of Theorem 15 and Corollary 16 in Caraballo and Real (2004), there exist positive constants \(\rho _V, \beta _1, \beta _2\) such that, for any bounded set \(D \subset C_H\) and for the above absorbing time \(T_{{\tilde{D}}}(t)\) corresponding to the set \(\{\mathcal {B}(t)\}_{t \in \mathbb {R}}\), it follows
for all \(s \geqslant T_D+1+r, t \in \mathbb {R}, \phi \in D\), and \(\theta _1, \theta _2 \in [-r, 0]\). In particular, the family \(\left\{ B_2(t)\right\} _{t \in \mathbb {R}}\), where \(B_2(t)=B_2=B_{C_V}\left( 0, \rho _V\right) \), is absorbing for the process \(U(\cdot , \cdot )\). Moreover, the family \(\left\{ B_S(t)\right\} _{t \in \mathbb {R}}\), where \(B_S(t)=B_{C_V}\left( 0, \rho _V\right) \times B_{L_V^2}\left( 0, h^{1 / 2} \rho _V\right) \), is absorbing for \(S(\cdot , \cdot )\).
Apparently, the above defined \(\left\{ B_2(t)\right\} _{t \in \mathbb {R}}\) is a family of bounded sets in \(C_V\), which is also (uniformly) absorbing for \({\tilde{U}}(\cdot , \cdot )\). Set \({\tilde{B}}_2=j\left( B_2\right) \), then, there exists \({\tilde{T}}_{{\tilde{B}}_2}=\) \(T_{B_2}+1+h>0\) such that \({\tilde{U}}(t, t-s) {\tilde{B}}_2 \subset B_2 \) for all \(t \in \mathbb {R}\), and all \(s \geqslant {\tilde{T}}_{{\tilde{B}}_2}\). Now, for each \(t \in \mathbb {R}\), consider the set
Thus, \(\left\{ B_3(t)\right\} _{t \in \mathbb {R}}\) is a family of uniformly bounded sets in \(C_V\) which is (uniformly) absorbing for \({\tilde{U}}(\cdot , \cdot )\). By similar techniques as those in the proof of Theorem 17 from Caraballo and Real (2004), each \(B_3(t)\) is relatively compact in X, then \(\left\{ \overline{B_3(t)}\right\} _{t \in \mathbb {R}}\) (where the closure is taken in X) is a family of compact absorbing sets in X for \({\tilde{U}}(\cdot , \cdot )\). Consequently, it is also a family of compact (uniform) absorbing sets for the process \(U(\cdot , \cdot )\) in X, which ensures the existence of the pullback attractors \(\{\mathcal {A}(t)\}_{t\in \mathbb {R}}\) for the process \(U(t, \tau )\). The uniqueness of these attractors holds since they are uniformly bounded. \(\square \)
Theorem 3.5
Let P be the m-dimensional projection on X defined by (3.28) and assume \(C_g^2+1-2(\mu _{1}+a)>0\). Then, we have
and
for any \(t \ge \tau \), and \(\varphi , \psi \) in \(\{\mathcal {A}(t)\}_{t\in \mathbb {R}}\), where \(\mu _1\), \(\mu _{m+1}\) and \(C_g\) are defined in (3.26) and \(\mathbf {Hypothesis\ A7}\), respectively.
Proof
For any two initial conditions \(\varphi , \psi \in C_V\), denote by \(u(\cdot )=u\left( \cdot ;\tau ,\varphi \right) \) and \(v(\cdot )=u\left( \cdot ; \tau , \psi \right) \) the corresponding solutions to (3.31) with initial time \(\tau \in \mathbb {R}\), respectively. Moreover, denote by \(y=\varphi -\psi \) and \(w(t)=u^\varphi (t)-u^\psi (t)\). It follows from (3.31) that
Denoting by \(w_m(t)=(I-P_m)w(t)\) and multiplying both sides of (3.31) by \(w^m(t)\) lead to
Keeping in mind that \((Aw^m, w^m)=-\Vert w^m\Vert \le -\mu _{m+1}|(I-P_m)w|\), then we have
Integrating both sides of (3.46) from \(\tau \) to t,
Consequently, for any \(t \geqslant \tau \), we have
The Gronwall lemma implies now, for any \(t \geqslant \tau \),
Assume that \(t \geqslant \tau +r\). Then, \(t+\theta \geqslant \tau \) for any \(\theta \in [-\tau , 0]\) and it follows from (3.26) that
and, thus,
This completes the proof of the second statement.
Now, we concentrate on the first part. Repeating the same procedure of the above proof but replacing \(w^m\) by w and \((Aw, w)=-\Vert w\Vert \le -\mu _{1}|w|\) gives rise to
By state decomposition, we have
Incorporating (3.51) and (3.52) into (3.53) leads to
Therefore, the first part holds, which completes the proof of the theorem. \(\square \)
Theorem 3.5 implies \(\left( \mathcal {H}_3\right) \) holds with \(M_1=2\textrm{e}^{-\mu _1 r}\left( C_g^2r+1\right) ^{1/2}, M_2=\textrm{e}^{-\mu _{m+1} r}\left( C_g^2r+1\right) ^{1/2}, M_3=0, \lambda _0=\frac{C_g^2+1-2(\mu _{1}+a)}{2}, \lambda _1=\frac{C_g^2+1-2(\mu _{m+1}+a)}{2}\). Hence, by Theorem 2.2, we have the following results about existence of a pullback exponential attractor \(\mathcal {M}(t)\) for the nonlinear dynamical system \(U(t, \tau )\) generated by (3.24).
Theorem 3.6
Let P be the finite-dimensional projection defined by (3.28), \(\mu _1\), \(\mu _{m+1}\) and \(C_g\) are defined in (3.26) and \(\mathbf {Hypothesis\ A7}\), respectively, and assumptions of Theorem 3.4 hold. Moreover, assume there exists \(\alpha >0\) such that \(\zeta :=\alpha e^{\frac{C_g^2+1-2(\mu _{1}+a)}{2}}+\textrm{e}^{-\mu _{m+1} r}\left( C_g^2r+1\right) ^{1/2}e^{\frac{C_g^2+1-2(\mu _{m+1}+a)}{2}}<1\). Then, (3.24) admits a pullback exponential attractor \(\mathcal {M}(t)\) whose fractal dimension has an upper bound
4 Retarded 2D Navier–Stokes Equations
In this section, we are concerned with the existence of pullback exponential attractors for the following delayed 2D Navier–Stokes equation on an open-bounded domain \(\Omega \subset \mathbb {R}^2\) with regular boundary \(\Gamma \),
Here, u is the velocity field of the fluid, \(\vartheta >0\) is the kinematic viscosity, f is a nondelayed external force field, p is the pressure, \(\tau \in \mathbb {R}\) is the initial time, g is another external force with some hereditary characteristics, and \(\phi \) the initial datum in the interval of time \([-r, 0]\) with r being a fixed positive number, representing the time delay. In the case \(t=\tau \), i.e., at the initial time, the initial velocity field is \(u(\tau ,x)=\phi (0,x)\). The investigation of retarded Navier–Stokes problems dates back to Caraballo and Real (2001, 2003), where the authors studied the existence and asymptotic behavior of solutions. In Caraballo and Real (2004), they further established the existence of pullback attractors. Recently, Qin and Su (2019) studied the Hausdorff dimension of Navier–Stokes–Voigt equations with a distributed delay, and they recast the equation in a Hilbert space and directly adopted the approach proposed in Constantin and Foias (1985) and Foias and Temam (1979). Here, we directly deal with the problem in the natural phase space, i.e., the Banach space. Moreover, we also consider the non-autonomous case.
Notation in this section are the same as those in Sect. 3.2 but with a general domain \(\Omega \) and the state in two-dimensional space. For instance, \(\mathcal {V}=\left\{ u \in \left( C_0^{\infty }(\Omega )\right) ^2: {\text {div}} u=0\right\} \) and H denotes the closure of \(\mathcal {V}\) in \(\left( L^2(\Omega )\right) ^2\). Other notation can be found in Sect. 3.2 and Caraballo and Real (2004). Let \(b:V \times V \times V\rightarrow \mathbb {R}\) be a trilinear form defined by
Set \(B: V \times V \rightarrow V^{\prime }\) by \(\langle B(u, v), w\rangle =\) b(u, v, w) and \(B(u)=B(u, u)\) \(\forall u, v, w \in V\). Denoting \(D(A)=H^2 \cap V\), then \(A u=-{\tilde{P}} \Delta u, \forall u \in D(A)\), (\({\tilde{P}}\) being the ortho-projector from \(\left( L^2(\Omega )\right) ^2\) onto \(\left. H\right) \). Hence, (4.1) can be recast in the following abstract form:
Denote \(X=C^0([-r, 0]; H), L_H=L^2(-r, 0; H)\) and \(L_V^2=L^2(-r, 0; V)\). By Douady and Oesterl (1980)[Theorem 2.3] and Caraballo and Real (2004)[Theorem 17,Corollary 16], we have the following existence of solutions as well as pullback attractors of (4.1), indicating \(\left( \mathcal {H}_1\right) \) and \(\left( \mathcal {H}_2\right) \) of Theorem 2.2 hold.
Lemma 4.1
Assume that \(f \in L_{\textrm{loc}}^2\left( \mathbb {R}; V^{\prime }\right) \), \(g: \mathbb {R} \times X \rightarrow \left( L^2(\Omega )\right) ^2\) such that hypotheses \(\mathbf {Hypothesis\ A4}\)–\(\mathbf {Hypothesis\ A7}\) hold. Then, the following statements hold.
(I) For each \(\tau \in \mathbb {R}\) and \(\phi \in C_V\), there exists a unique solution u to (5.2) which belongs to the space \( C^0([\tau -r,+\infty ); V)\). (II) Let \(U(t, \tau ):C_H \rightarrow C_H\) be defined by \(U(t, \tau )\phi =u_t(\cdot , \tau , \phi )\) for all \(\tau \leqslant t\) and \(\phi \in C^0([\tau -r, t]; H)\) and \(\mu _1\) is the first eigenvalue of the operator A. If \(m_0>0\) and \(\vartheta \mu _1>C_g\), then there exist a positive constant \(\rho _V\) and a unique uniformly bounded pullback attractor \(\left\{ \mathcal {A}(t)\right\} _{t \in \mathbb {R}}\), which is inside \(B_{C_V}(0, \rho _V)\), the absorbing ball in \(C_V\) with center 0 and radius \(\rho _V\).
Let \(V_m\) and P being defined by (3.25) and (3.28) in Sect. 3.2, then we have the following results concerning the squeezing property.
Theorem 4.1
Let P be defined by (3.28), \(\mu _1, \rho _V\) be given in Lemma 4.1. Assume that \(\vartheta \mu _1>C_g\) and there exists \(c_0^{ \prime }>0\) such that \(C_g^2+c_0^{ \prime }\rho _V-\vartheta \mu _{1}+1>0\). Then, we have
and
for any \(t \ge \tau \), and \(\varphi , \psi \) in \(\mathcal {A}(t)\), where \(\mu _1\), \(\mu _{m+1}\) and \(C_g\) are defined in (3.26), (3.27) and \(\mathbf {Hypothesis\ A7}\), respectively.
Proof
For any two initial conditions \(\varphi , \psi \in \mathcal {A}(t)\), denote by \(u(\cdot )=u\left( \cdot ;\tau ,\varphi \right) \) and \(v(\cdot )=u\left( \cdot ; \tau , \psi \right) \) the corresponding solutions to (4.2) with initial time \(\tau \in \mathbb {R}\), respectively. Moreover, denote \(w(t)=u(t)-v(t)\). It follows from (4.2) that
Multiply both sides of (4.2) by w(t) and take into account that \(B(u(t))-B(v(t))=B(w(t),u(t))+B(v(t),w(t))\), then
Keeping in mind that \((Av,v)=-\Vert v\Vert \le -\mu _{1}|v|\), we have
and therefore, noticing that \(u(t)\in \mathcal {A}(t)\), we have \(\Vert u(t)\Vert ^2\le \rho _V\), and
Consequently,
The Gronwall lemma implies now for any \(t \geqslant \tau \),
Assume now that \(t \geqslant \tau +r\), then \(t+\theta \geqslant \tau \) for any \(\theta \in [-\tau , 0]\) and it holds
Thus, by (3.30), we have
implying that
Now, we concentrate on the first part. By (3.29) and (4.11), we have
Therefore, the first part holds, which completes the proof of the theorem. \(\square \)
Theorem 3.5 implies \(\left( \mathcal {H}_3\right) \) holds with \(M_1=e^{-\mu _{1}r}(C_g^2r+1)^{1/2}, M_2= e^{-\mu _{m+1}r}(C_g^2r+1)^{1/2}, M_3=0, \lambda _0=\lambda _1=\frac{\left( C_g^2+c_0^{ \prime }\rho _V-\vartheta \mu _{1}+1\right) }{2}\). Hence, by Theorem 2.2, we have the following results about existence of a pullback exponential attractor \(\mathcal {M}(t)\) for the nonlinear evolution process \(U(t,\tau )\) generated by (4.2).
Theorem 4.2
Let P be the finite-dimensional projection defined by (3.28), \(\mu _1\), \(\mu _{m+1}\) and \(C_g\) defined in (3.26), (3.27) and \(\mathbf {Hypothesis\ A7}\), respectively, and assumptions of Lemma 4.1 and Theorem 4.1 hold. Moreover, assume there exists \( \alpha >0\) such that \(\zeta :=(\alpha +e^{-\mu _{m+1}r}(C_g^2r+1)^{1/2}) e^{\frac{(C_g^2+c_0^{ \prime }\rho _V-\vartheta \mu _{1}+1)}{2}}<1\). Then, (4.2) admits a pullback exponential attractor \(\mathcal {M}(t)\) with fractal dimension satisfying
Remark 4.1
In [Theorem 6.2, Qin and Su (2019)], the authors investigated dimensions of global attractors of the following 2D Navier–Stokes–Voigt equations with a distributed delay
by recasting (4.16) into a Hilbert space and adopting the method established in Constantin and Foias (1985). The upper bounds of the Hausdorff and fractal dimensions of \(\mathcal {A}\) they gave is
which do not depend on the time delay. In the case \(\alpha =0\), (4.16) degenerates to (4.1). However, in this case, (4.17) is not well-defined since zero appears in the denominator, which means that the method in Constantin and Foias (1985) may be ineffective for obtaining dimensions of (4.1). Moreover, compared with (4.17), our results depend on the time delay \(\tau \), which shows the characteristic of the equation.
5 Retarded Semilinear Wave Equations
This section is dedicated to dimension estimations of pullback attractors for the following retarded semilinear wave equation defined on an open bounded domain \(\Omega \subset \mathbb {R}^n, n \ge 1\), with a smooth boundary \(\partial \Omega =\Gamma \)
Here, \(\beta \) is a positive constant, \(f+g\left( t, u_t\right) \) is the source intensity which may depend on the history of the solution, \(\phi \) is the initial datum on the interval \([\tau -r, \tau ]\) where \(r>0\), and, as usual, \(u_t\) is defined for \(\theta \in [-r, 0]\) as \(u_t(\theta )=u(t+\theta )\) as well. The existence of global unique solutions and pullback attractors of (5.1) have been studied in Caraballo et al. (2004) and Garrido-Atienza and Real (2003), respectively. In the present work, we go a step further to establish explicit dimensions estimation of the pullback attractors. The notation in this subsection have the same meaning as those of Sect. 3.2 but with the domain being \(\Omega \). Thus, problem (5.1) can be written as a second order differential equation in H.
We first introduce more notations. Let Y be H or V, denote by \(C_Y\) the space \(C^0([-\tau , 0]; Y)\) with the sup-norm, i.e., \(\Vert \phi \Vert _{C_Y}=\sup _{\theta \in [-\tau , 0]}\Vert \phi (\theta )\Vert _Y\), for \(\phi \in C_Y\). Given another Banach space \(\left( Z,\Vert \cdot \Vert _Z\right) \) such that the injection \(Y \subset Z\) is continuous, we denote by \(C_{Y, Z}\) the Banach space \(C_Y \cap C^1([-\tau , 0]; Z)\) with the norm \(\Vert \cdot \Vert _{C_{Y, Z}}\) defined by
We will use the spaces \(C_{D(A)}, C_V, X, C_{V, H}\) and \(C_{D(A), V}\) in our analysis. Apart from \(\mathbf {Hypothesis\ A4}\)–\(\mathbf {Hypothesis\ A7}\), we impose one more hypothesis on the function \(g: \mathbb {R} \times X \rightarrow H\).
\(\mathbf {Hypothesis\ A8}\) \(g \in C^1\left( \mathbb {R} \times X; H\right) \), and there exists \(C>0\) such that, for any \((t, \xi ) \in \mathbb {R} \times X\) the Fréchet derivative \(\delta g(t, \xi ) \in \mathcal {L}\left( \mathbb {R} \times X, H\right) \) satisfies
By the results in Garrido-Atienza and Real (2003) and Caraballo et al. (2004), we have the following ones.
Lemma 5.1
Assume that \(f \in L_{l o c}^2(\mathbb {R}, H), \phi \in C_{V, H}\) and g satisfies \(\mathbf {Hypothesis\ A4}\)–\(\mathbf {Hypothesis\ A7}\). Then, for any \(\tau \in \mathbb {R}\), there exists a unique solution \(u(\cdot )=u(\cdot ; \tau , \phi )\) to problem (5.1) such that \(u \in \) \(C^0([\tau -r, \infty ); V) \cap C^1([\tau -r, \infty ); H)\). If in addition \(f^{\prime } \in L^2(\tau , T; H)\) for all \(T>0, \phi \in C_{D(A)}\), \(\phi ^{\prime } \in C_V\), then
In addition, suppose that \(\mathbf {Hypothesis\ A8}\) holds and \(2 \sqrt{2} \mu _1^{-1 / 2} C_g<\min \left\{ \frac{\beta }{4}, \frac{\mu _1}{2 \beta }\right\} \). Then, there exists a family \(\{B(t)\}_{t \in \mathbb {R}}\) of bounded sets in \(C_{V, H}\) which is uniformly pullback (and forward) absorbing for the process \(U(\cdot , \cdot )\).
Let \(V_m\) and \(P_m\) be defined in Sect. 3.2. Replace V and H by \(V_m\) and \(H_m\) gives the definiton of \(C_{V_m, H_m}\), which is a finite-dimensional subspace of the Banach space \(C_{V, H}\). Define \({\tilde{P}}_m: C_{V, H}\rightarrow C_{V_m, H_m}\). Then, we have the following results about the squeezing property.
Theorem 5.1
Let P be the finite-dimensional projection \({\tilde{P}}_m\) on \(C_{V, H}\). Assume that \(\mathbf {Hypotheses\ A4-A7}\) hold for any \(\tau \leqslant t\) with \(m_0>0\), \(\mathbf {Hypothesis\ A8}\) holds and \(2 \sqrt{2} \mu _1^{-1 / 2} C_g<\min \left\{ \frac{\beta }{4}, \frac{\mu _1}{2 \beta }\right\} \). Then there exists a constant \(\gamma >0\) such that
and
for any \(t \ge \tau +r\), and \(\varphi , \psi \) in \(\mathcal {A}(t)\), where \(\mu _1\), \(\mu _{m+1}\) and \(C_g\) are defined in (3.26), (3.27) and \(\mathbf {Hypothesis\ A7}\), respectively.
Proof
Let \(\phi , \psi \in C_{V, H}\) be two initial data for our problem (5.2), and let \(\tau \in \mathbb {R}\) be an initial time. Denote by \(u(\cdot )=u(\cdot ; \tau , \phi )\) and \(v(\cdot )=u(\cdot ; \tau , \psi )\) the corresponding solutions to (5.2). Then, it follows from Caraballo et al. (2004)[Lemma 3.1], there exists a constant \(\gamma >0\) which does not depend on the initial data and time, such that for all \(t \ge \tau +r\)
implying that
Thus, by (3.30) and (3.29), we have
and
Therefore, the first part holds, which completes the proof of the theorem. \(\square \)
Theorem 5.1 implies \(\left( \mathcal {H}_3\right) \) holds with \(M_1=e^{-\mu _{1}r}\left( 1+\mu _1^{-1} C_g^2 r\right) ^{\frac{1}{2}}, M_2=e^{-\mu _{m+1}r}\left( 1+\mu _1^{-1} C_g^2 r\right) ^{\frac{1}{2}}, M_3=0, \lambda _0=\mu _1=\frac{\gamma }{2}\) Hence, by Theorem 2.2, we have the following results about existence of a pullback exponential attractor \(\mathcal {M}(t)\) for the nonlinear evolution process \(U(t, \tau )\) generated by (5.2).
Theorem 5.2
Let P be the finite-dimensional projection defined by (3.28), \(\gamma \) and \(C_g\) are defined in (3.26), (3.27) and \(\mathbf {Hypothesis\ A7}\), respectively, and assumptions of Lemma 5.1 and Theorem 5.1 hold. Moreover, assume there exists \( \alpha >0\) such that \(\zeta :=(\alpha +e^{-\mu _{m+1}r}\left( 1+\lambda _1^{-1} C_h^2 r\right) ^{\frac{1}{2}})e^{\frac{\gamma }{2}}<1\). Then, (5.2) admits a pullback exponential attractor \(\mathcal {M}(t)\) whose fractal dimension has an upper bound
6 Summary
In this paper, we established a new framework to construct exponential attractors for infinite-dimensional non-autonomous dynamical systems in Banach spaces with explicit fractal dimension. To our best knowledge, there are two kinds of methods can be used to investigate exponential attractors of infinite-dimensional dynamical systems, the first one is due to Eden, Foias, Nicolaenko and Temam Eden et al. (1994), which depends on squeezing property and phase space decomposition and is established for PDEs in Hilbert spaces. Here, we extended this method to non-autonomous case in Banach spaces. The other well-known method, which is also effective for constructing exponential attractors in Banach spaces, was firstly proposed by Efendiev, Miranville and Zelik which depends on smoothing property and compact embedding of the systems between two spaces (Efendiev et al. 2000; Efendiev and Zelik 2008). Compared with their woks, the method here does not need the smoothing property and the entropy number of the embedding between two spaces but requires to some extend appropriate conditions on the spectrum gap and the Lipschitz constant of the nonlinear term. As pointed out in Constantin and Foias (1985), the dimensions estimation by squeezing property method may not be optimal and the more accurate method should be the Lyapunov exponents method which depends on computing traces of some linear operators generated by the linearization of the equations, requiring quasi-differentials of the underlying systems. Nevertheless, the method requires the smooth inner product of the Hilbert space geometric structure. How to extend this method to Banach spaces deserves much more effort since the Lyapunov exponents for evolution equations in Banach spaces may be not easy to obtain.
The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on inner characteristics of the studied equations. The method shows a wide applicability to infinite-dimensional dynamical systems generated by partial functional differential equations, including the retarded reaction–diffusion equations, the retarded 2D Navier–Stokes equations and the retarded semilinear wave equations. They maybe also available for investigating topological dimensions of attractors for neutral partial functional differential equations, the infinite delay case as well as some other evolution equations with certain squeeze properties in Banach spaces.
In the applications, we only consider partial functional differential equations on bounded domain. Actually, there are many real world process evolution on infinite domain, such as the mature population of species living in an infinite habitat. In such a scenario, the Laplace operator has a continuous spectrum, \(H^1(\mathbb {R}^n)\) is not compactly embedded in \(L^2(\mathbb {R}^n)\) and the solution semiflow do not have absorbing sets that are compact in the original topology, causing the method developed here no longer effective and new techniques should be established. This will be studied in an upcoming paper.
The definition of pullback attractors generally means which states in the infinite past will evolute to the given present state. Indeed, from causality perspective, we are more interested about what present states will evolute to in the future. The effective tools for describing the future evolution of non-autonomous system are the uniform attractor and uniform exponential attractors, which will also be studied in the near future.
Generally, random effects are omnipresent in mathematical modelings. Therefore, one anther question is, whether there are exponential attractors with explicit fractal dimension for partial functional differential equations perturbed by random effect, i.e., the stochastic partial functional differential equations(SPFDEs)? Indeed, even under what conditions do SPFDEs generate random dynamical systems have not been perfectly tackled needless to say the state decomposition and exponential dichotomy. This problem also deserves much effort in the future.
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Acknowledgements
The authors would like to thank the anonymous referee for his/her valuable comments, which lead to an improvement in our presentation. This work was jointly supported by the Scientific Research Fund of Hunan Provincial Education Department (23C0013), China Scholarship Council(202008430247). The research of T. Caraballo has been partially supported by Spanish Ministerio de Ciencia e Innovación (MCI), Agencia Estatal de Investigación (AEI), Fondo Europeo de Desarrollo Regional (FEDER) under the project PID2021-122991NB-C21. This work was completed when Wenjie Hu was visiting the Universidad de Sevilla as a visiting scholar, and he would like to thank the staff in the Facultad de Matemáticas for their hospitality and thank the University for its excellent facilities and support during his stay.
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Hu, W., Caraballo, T. Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations. J Nonlinear Sci 34, 27 (2024). https://doi.org/10.1007/s00332-023-10003-5
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DOI: https://doi.org/10.1007/s00332-023-10003-5
Keywords
- Pullback exponential attractors
- Non-autonomous
- Partial functional differential equations
- Squeezing property
- Fractal dimension
- Delay