Abstract
The aim of this paper is to construct invariant manifolds for a coupled system, consisting of a parabolic equation and a second-order ordinary differential equation, set on \(\mathbb {T}^3\) and subject to periodic boundary conditions. Notably, the “spectral gap condition" does not hold for the system under consideration, leading to the use of the spatial averaging principle, together with the graph transform method. This approach facilitates the construction of the relevant invariant manifold, characterized by attributes such as Lipschitz continuity, local invariance, infinite dimensionality, and exponential tracking, thus mirroring the properties traditionally associated with a classical global manifold.
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1 Introduction
From a physical point of view, the energy of a dissipative dynamical system diminishes over time. Consequently, the asymptotic behavior of such systems can often be described by global attractors, invariant manifolds, inertial manifolds, and center manifolds [37, 40]. When studying the long-time behavior of dynamical systems, the complexity of infinite-dimensional systems often poses challenges. To address this, attempts are frequently made to transform infinite-dimensional systems into finite-dimensional counterparts, leveraging a finite number of degrees of freedom to represent the entire system [37, 40, 44].
Encouragingly, in certain scenarios, the dynamics of infinite-dimensional dissipative dynamical systems align with finite-dimensional structures, such as finite-dimensional attractors, finite-dimensional invariant manifolds, and inertial manifolds [37]. The concept of invariant manifolds was initially introduced by Hadamard [28]. These objects significantly contribute to reducing the complexity inherent to nonlinear dynamical systems. Typically, an invariant manifold is conceived as a perturbation of the invariant subspace associated with equilibria [43]. Comprehensive studies of invariant manifolds for both deterministic and random dynamical systems can be found in references [1, 6, 17,18,19, 22,23,24, 42].
A global invariant manifold adequately captures most of the dynamics contained in a dissipative infinite-dimensional dynamical system, since such a manifold contains the global attractor. Extensive research has been dedicated to global manifolds, with seminal contributions that can be found in references [6, 10, 15, 31, 44]. In parallel, abundant results exist regarding local invariant manifolds. Studies pertaining to local invariant manifolds associated with the vicinity of equilibria and ground state solutions can be found in [2, 3, 29, 38]. Moreover, manifolds related to solitons have been explored in [20, 30]. Such manifolds provide a precise description of local dynamical behaviors.
To facilitate the reduction of an infinite-dimensional dynamical system to a finite form, Foias, Sell, and Temam [15] proposed the concept of an inertial manifold. An inertial manifold is defined as a finite-dimensional Lipschitz invariant manifold that exponentially attracts the system’s solutions [15, 31] and encompasses the global attractor. The existence of an inertial manifold for a specific partial differential equation (PDE) yields that the equation can be reduced to a set of finite-dimensional ordinary differential equations (ODEs), named inertial form. Consequently, the asymptotic behavior of the original equation can be mirrored by the long-time behavior of the corresponding inertial form [4, 10, 14, 21, 33, 41].
Unfortunately, we usually prove the existence of an inertial manifold under the “spectral gap condition" [15]. In fact, in special cases, if we consider an equation in \(\mathbb {T}^{m}=(-\pi ,\pi )^m\) with periodic boundary conditions, we have, for the eigenvalues \(\lambda _{n} \) of the Laplace operator,
C is a positive constant, so that, when \(m=1\),
The spectral gap condition “\(\lambda _{n+1}-\lambda _{n} \>2\,L\)”(L is the Lipschitz constant of a nonlinearity) can be easily satisfied if n is large enough [37]. However, if we take \(m=2\), then \(\lambda _{n+1}-\lambda _{n}=C\), so that the spectral gap condition fails when L is large.
There is a strong interest in performing a finite-dimensional reduction on a global attractor with finite fractal dimension. Indeed, it is easier to prove the existence of the global attractor than that of an inertial manifold. Unfortunately, this may lead to a poorly behaved system of ODEs, namely, Hölder continuous rather than Lipschitz continuous [44]. Consequently, the reduction via an invariant manifold becomes necessary.
The concepts of invariant manifolds and inertial manifolds possess a strong correlation with the Lyapunov stability [36]. Historically, an inertial manifold could be deduced from Sacker’s equation [8], elliptic regularity theory [13], among other methodologies. Of these techniques, the Hadamard’s graph transform, a geometric method, and the Lyapunov-Perron method, an analytic approach, are the most prevalent. Both methods are strongly related to the "spectral gap condition". Within the Lyapunov-Perron approach, the Lyapunov function is characterized by the difference between the “finite-dimensional part" and the “infinite-dimensional part" [21]. Subsequently, the cone property is derived, wherein the “spectral gap condition" or other techniques come into play. As for the Hadamard’s graph transform, the process starts with the flat manifold at \(t=0\), transitioning to the limit as \(t\rightarrow \infty \) and culminating with the construction of the desired manifold [14, 31].
Clearly, obtaining the “spectral gap condition" becomes challenging for a space dimension \(m\ge 2\) for periodic domains or when the Lipschitz constant L is large. Under these circumstances, alternative approaches become essential. Establishing the existence of an inertial manifold without relying on the spectral gap condition warrants attention. Numerous contributions have been made in this direction [22, 24, 31]. A predominant strategy involves employing the spatial averaging principle, initially introduced in reference [31] to address reaction-diffusion equations in higher dimensions.
In a recent work, the authors in [16] proved the existence of an inertial manifold for the incompressible hyperviscous Navier–Stokes equations on the 2D or 3D torus with periodic boundary conditions, with a hyperdissipation range \( \beta \ge \frac{3}{2}\) in the fractional Laplacian \((-\bigtriangleup )^{\beta }\). In [23], an inertial manifold for the modified Leray-\(\alpha \) model with periodic boundary conditions on the 3D torus is constructed. Incorporating the insights from the aforementioned studies, the authors in reference [27] established the existence of an inertial manifold for the critical modified-Leray model on the 3D torus. The technical approach employed involved using the Leray–Helmholtz orthonormal projector to truncate the equation within a regular absorbing ball. Subsequent operations are facilitated by the “spatial averaging principle" as an alternative to the classical spectral gap condition. It is worth mentioning that the work in [31] was first generalized to a coupled system [39]; this paper was concerned with a system consisting of a parabolic PDE and a first-order ODE in a rectangular domain \((0,2\pi /a_1)\times (0,2\pi /a_2)\) in \(\mathbb {R}^{2}\) or a cubic domain \((0,2\pi /a_1)\times (0,2\pi /a_2)\times (0,2\pi /a_3)\) in \(\mathbb {R}^{3}\), \((\frac{a_i}{a_j})^2\in \mathbb {Q}\).
In this paper, we consider the following coupled system on the torus \(\mathbb {T}^3=(-\pi ,\pi )^3\) with periodic boundary conditions:
In addition, \(\Delta \) is the Laplace operator, \(\beta >0\) and f(v) and \(\varphi (v)\) are two nonlinearities. The problem is inspired by [12] where the authors considered the model in a bounded domain in \(\mathbb {R}^3\) with Neumann boundary conditions. There, in particular, the existence of a finite-dimensional attractor is obtained. System (1.1) was introduced in [26] in order to describe the development of a forest ecosystem, and then further studied by many authors analytically and numerically [5, 34]. Such models, defined on bounded domains with Dirichlet, Neumann or periodic boundary conditions, have been employed in other fields [25, 32]. Typically, system (1.1) is called a partly dissipative reaction-diffusion system; such systems have received a lot of attention in recent years.
The dynamical behavior of system (1.1) is strongly influenced by the monotonicity of the ODE. In instances of non-monotonicity, asymptotic discontinuities emerge, and a smooth spatial profile of initial values may be absent due to the ODE instability. Conversely, in the monotonic scenario, an initial instability is mitigated, leading to the asymptotic compactness of the solution semigroup and the presence of smooth finite-dimensional attractors [11, Chapter 11].
Moreover, system (1.1) can be rewritten in the following equivalent form:
Here, \(\beta , g, h\) are positive constants (\(g<h\) in this paper) and r(w) is a nonlinearity. Actually, if we substitute the second equation of (1.2), which can be expressed as \(v=\frac{\frac{dw}{dt}+hw}{g}\), into the first equation of (1.2), we can obtain (1.1) immediately. The global well-posedness has been investigated in [12]. The global attractor of the system when the nonlinearity f(v) in the ODE is monotone was obtained in [11, Chapter 11] (see also [12]). Nevertheless, there exists no result concerning invariant and inertial manifolds that offer a comprehensive reduction of system (1.1). When juxtaposing the available results for invariant manifolds of PDE-ODE coupled systems, it is clear that the two phase spaces examined in this model are Banach spaces, and the ODE is of second-order. Our approach in this paper is flexible and can be used for many other parabolic PDE-ODE coupled systems, see e.g. [7, 25, 32] and reference therein.
Structure of the paper:
In order to get the existence of invariant manifolds for (1.1) when f(v) is monotone, we need several steps as follows.
In Sect. 2, basic concepts related to inertial manifolds are introduced. These serve as critical applications of the “spectral gap condition," ensuring the existence of an inertial manifold, as detailed in Theorems 2.1 and 2.2.
Section 3 focuses on addressing the model problem presented in (2.1) by identifying sufficient conditions that ensure the existence of an inertial manifold. Notably, the spatial averaging principle, first established in [31] and later extended by the author in [44], is employed to handle PDE models without the spectral gap condition.
In Sect. 4, the abstract differential system corresponding to problem (1.1) is examined. Under appropriate conditions, the well-posedness of the problem is outlined in Sect. 4.1. Subsequently, it becomes necessary to extend both the uniform cone condition and the spatial averaging method (as discussed in Sect. 3) to this abstract differential system. These extensions are detailed in Sects. 4.2 and 4.3.
In Sect. 5, the theory formulated in Sect. 4 is applied to the coupled PDE-ODE equations (1.1). The system is confined within an absorbing ball, as detailed in Sect. 5.1. A subsequent verification of all crucial conditions related to the abstract system is provided in Sect. 5.2. Consequently, the existence of an infinite-dimensional invariant manifold for (1.1) is presented as the graph of a Lipschitz continuous function \(\Psi \):
here \(\mathcal {P}\) is finite-dimensional, (v, w) is the solution of the ODE, and \(\Psi :\mathcal {P} \times L^{\infty }(\mathbb {T}^3)\times L^{\infty }(\mathbb {T}^3)\rightarrow \mathcal {Q}\) is Lipschitz continuous. Moreover, the manifold is exponentially attracting and \(\mathcal {Q}=H^1\setminus \mathcal {P}\). As is typical, this infinite-dimensional manifold remains invariant under the solution semigroup associated with (1.1). In the vicinity of the global attractor, this manifold encompasses all its non-trivial dynamics.
2 Preliminaries
Let H be a Hilbert space and consider an equation on H,
where \(u=u(x,t),(x,t)\in \Omega \times [0,\infty )\), \(\Omega \subset \mathbb {R}^n\) is a bounded domain with smooth boundary, \(F:H \rightarrow H\) is a nonlinearity and A is a linear self-adjoint positive operator with compact inverse. We assume that A possesses a complete orthonormal system of eigenvectors \(\{e_{j}\}^{\infty }_{j=1}\) in H that corresponds to eigenvalues \(\lambda _{j}\), where \(\lambda _{j} \rightarrow \infty \) as \(j \rightarrow \infty \) and
The norm and inner product of H are denoted by \(\Vert \cdot \Vert \) and \((\cdot ,\cdot )\) respectively. Hence, any \(u\in H\) can be expressed as
Recall the Parseval equality:
The norms in the spaces \(H^s=D(A^{\frac{s}{2}}),s\in \mathbb {R}\), are given by
Define projection operators as:
where \(u_{j}=(u,e_{j})\) are the Fourier coefficients of u.
Set \(H_+=P_{N}H\) and \(H_-=Q_{N}H\). These spaces are invariant under the operator A, resulting in:
Definition 2.1
(Global attractor) [40] A set \(\mathcal {A} \subset H\) is called a global attractor if it enjoys the following properties:
(1) \(\mathcal {A}\) is an invariant set, that is \(S(t)\mathcal {A}=\mathcal {A}, \forall t \ge 0\);
(2) \(\mathcal {A}\) possesses an open neighborhood \(\mathcal {B}\) such that \(\forall u_{0}\in \mathcal {B}\),
We call \(\mathcal {A}\) the global attractor for the semigroup \(\left\{ S(t)\right\} _{t\ge 0}\) if \(\mathcal {A}\) is the compact attractor that attracts all bounded set of H.
Definition 2.2
(Absorbing set) [40] Let \(\mathcal {C}\) be a subset of H and \(\mathcal {U}\) be an open set containing \(\mathcal {C}\). We say that \(\mathcal {C}\) is absorbing in \(\mathcal {U}\) if the orbit of any bounded set of \(\mathcal {U}\) enters \(\mathcal {C}\) after a certain time (which may depend on the set):
We call \(\mathcal {C}\) an absorbing set that absorbs the bounded sets of \(\mathcal {U}\).
Definition 2.3
(Inertial manifold) [15] A subset \(\mathcal {M}\subset H\) corresponding to the semigroup S(t) is called an inertial manifold for a dynamical system in H provided that it has the following properties:
(1) \(\mathcal {M}\) is invariant with respect to the semigroup S(t), i.e. \(S(t)\mathcal {M}=\mathcal {M},\forall t\ge 0\);
(2) \(\mathcal {M}\) is a finite-dimensional Lipschitz manifold, i.e. there exists a Lipschitz continuous function \(\Phi :P_{N}H\rightarrow Q_{N}H\) such that
(3) (exponential tracking property) there exist constants \(C,\alpha \>0\) such that \(\forall u_{0}\in \)H, there exists \(v_{0}\in \mathcal {M}\) such that
Significant findings regarding the existence of an inertial manifold for equation (2.1), established based on the “spectral gap condition", are presented below [44]. Indeed, under these circumstances, the nonlinear system may be viewed as a minor perturbation of its linear counterpart. The invariant manifold \(H_+\) from the unperturbed equations remains present in the perturbed system.
Theorem 2.1
[44] Under the assumptions on A and assuming that F in (2.1) is Lipschitz continuous with a Lipschitz constant L and that for some \(N\in \mathbb {N}\), the spectral gap condition is satisfied:
then there exists an N-dimensional inertial manifold.
Set \(u_+=P_Nu\) and \(u_-=Q_Nu\). Next we decompose equation (2.1) as follows
where \(F_+=P_NF, F_-=Q_NF\). If an inertial manifold exists, we project system (2.1) onto the inertial manifold and we have \(u_-(t)=\Phi (u_+(t))\), so that the dynamical behavior of the whole system can be determined by the \(N-\)dimensional system of ODEs
which is often called the inertial form of equation (2.1).
The following theorem extends the above one.
Theorem 2.2
[44] Suppose that F in (2.1) satisfies
where \(H^{\beta }=D(A^{\frac{\beta }{2}})\), \(-2< \beta \le 0\).
If there exists \( N \in \mathbb {N}\) such that the spectral gap condition is valid:
then (2.1) possesses an N-dimensional inertial manifold.
The subsequent lemma pertains to the “spatial averaging principle." This principle dictates the conditions under which its application is appropriate.
Lemma 2.1
[31] Let \(a=(a_{1}, a_{2})\in \mathbb {R}^{2}\) with each \(a_{j}>0\), and consider the points \(ak=(a_{1}k_{1}, a_{2}k_{2})\), where \(k=(k_{1}, k_{2})\in \mathbb {Z}^{2}\). Then \(\exists \ \xi \>0\) such that \(\forall \ \kappa >1, d\>0\), there exists an arbitrary large \(\lambda \) satisfying the following conditions:
(1) Whenever \(|ak|^{2}, |al|^{2}\in (\lambda -\kappa , \lambda +\kappa ]\), with \(\kappa , l\in \mathbb {Z}^{2}\), one has either \(k=1\) or \(|ak-al|\ge d\);
(2) \(|ak|^{2}\notin (\lambda -\frac{1}{2}\xi , \lambda +\frac{1}{2}\xi )\), for each \(k\in \mathbb {Z}^{2}\).
For \(a=(1, 1,1)\in \mathbb {R}^{3}\), the corresponding result holds for lattice points \(k\in \mathbb {Z}^{3}\) with \(\xi =1\).
3 Inertial Manifolds Without the Spectral Gap Condition
The analysis starts with an abstract PDE, for which the existence of an inertial manifold is established. This approach draws inspiration from references [31, 44]. Define the abstract PDE on a subset \(\Omega \subset \mathbb {R}^{n}\), omitting the spectral gap condition. Specifically, the difference \(\lambda _{n+1}-\lambda _{n} \) does not exceed 2L, where L represents the Lipschitz constant of the nonlinearity in equation (2.1).
Three projection operators, corresponding to the low, intermediate, and high Fourier modes, are defined as follows:
where \(u_{j}=(u,e_{j})\) and \(k<\lambda _{N}\).
Theorem 3.1
(Uniform Cone Condition) Let \(u_{1}, u_{2}\) be two solutions of (2.1) in H, set \(w=u_{1}-u_{2}\), \(p=P_{N}w\), \(q=Q_{N}w\), and assume that \(F:H\rightarrow H\) is Gateaux differentiable, with \(\forall u\in H\), \(\Vert F'(u)\Vert _{\mathcal {L}(H, H)}\le L\). If there exists \(N\in \mathbb {N}\) such that \(4+6\delta <\lambda _{N+1}-\lambda _{N}\le 2\,L\), \(k>\frac{9\,L^{2}+1}{4}+2+3\delta \), \(\delta >0\), and the following spatial averaging condition holds:
then we can define \(V(t)=\Vert q(t)\Vert ^{2}-\Vert p(t)\Vert ^{2}\) and there exists \(\mu >0\) making the following uniform cone condition valid:
Proof
For \(w=u_{1}-u_{2}\), then we have
Taking the inner product with p and q, we get
Subtracting the two equations above, we have
Set \(\alpha =\frac{\lambda _{N+1}+\lambda _{N}}{2}\). From the mean value theorem for integrals, there holds
so that we obtain
Actually we need estimate each term on the right-hand side of (3.2).
Since \(\Vert w\Vert ^{2}=\Vert q\Vert ^{2}-\Vert p\Vert ^{2}\), and from the definition, we know that
By a simple computation, we can obtain
Analogously, let \(p=P_{k,N}w+R_{k,N}p\). Then
so that
Since \(q=Q_{k,N}w+R_{k,N}q\), and
we can get
Owing to the following two inequalities:
we substitute (3.6) into (3.5) to get
In order to deal with the nonlinearity, we first estimate the term \((F'(u)w,q-p)\). \(\forall u\in \) H, we have
Notice that
Hence we get
Owing to \(\Vert F'(u)\Vert _{\mathcal {L}(H,H)}\le L\), by using the Cauchy inequality with \(\epsilon \), we obtain
Similarly,
and
Besides, applying the spatial averaging condition, we get
and
A substitution of (3.9)–(3.14) into (3.8) gives
Then we can get the desired estimate by substituting (3.3), (3.4), (3.7), (3.15) into (3.2),
Combining the above with \(\lambda _{N+1}-\lambda _{N}\le 2L \), we end up with
and a simplification of (3.16) leads to
Since \(\delta >0\), \(\alpha >0\), we can deduce that
Finally we arrive at
Here \(\mu =\frac{\lambda _{N+1}-\lambda _{N}}{2}-2-3\delta >0\), making the uniform cone condition valid. This completes the proof. \(\square \)
We next get the strong squeezing property through the uniform cone condition.
Theorem 3.2
(Strong Squeezing Property) Let \(u_{1}, u_{2}\) be two solutions of (2.1) in H. Set \(w=u_{1}-u_{2}\), \(p=P_{N}w\), \(q=Q_{N}w\). If there exist \( \alpha , \mu \>0\) such that the uniform cone condition holds, in other words, \(\forall t\ge 0\), \(\frac{d}{dt}V(t)+2\alpha V(t)\le -\mu \Vert w(t)\Vert ^{2}\), where \(V(t)=\Vert q(t)\Vert ^{2}-\Vert p(t)\Vert ^{2}\), then the following strong squeezing property hold:
(1) (cone invariance property) Provided that \(V(0)\le 0\), we have \(\forall t\ge 0\), \(V(t)\le 0\); on the other hand, if \(T>0\) makes \(V(T)>0\), then \(\forall t \in [0,T]\), \(V(t)>0\);
(2) (decay property) If \(V(T)>0\) holds for \(T>0\), we can deduce that for a positive constant C,
Proof
Since \(\forall t\ge 0\), \(\frac{d}{dt}V(t)+2\alpha V(t)\le -\mu \Vert w(t)\Vert ^{2}\), by the Gronwall inequality, we have
which implies the cone invariance directly, so that we just need to prove the decay property.
Take the inner product of (3.1) with w. Together with \(\Vert F(u_{1})-F(u_{2})\Vert \le L\Vert u_{1}-u_{2}\Vert \), we have
For \(\Vert A^{\frac{1}{2}}w(t)\Vert ^{2} \ge 0\), we obtain
Then define \(V_{\epsilon }(t)=\epsilon \Vert w(t)\Vert ^{2}+V(t)\), so that
Set \(\epsilon =\frac{\mu }{\alpha +2L}\). Then by Gronwall’s inequality, we have \(\forall t>0\), \(V_{\epsilon }(t)\le V_{\epsilon }(0)e^{-\alpha t}\).
Since \(V(t)=\Vert q(t)\Vert ^{2}-\Vert p(t)\Vert ^{2}\le \Vert q(t)\Vert ^{2}+\Vert p(t)\Vert ^{2}=\Vert w(t)\Vert ^{2}\), we see that
that is
The decay property is proved, which completes the proof. \(\square \)
Through the cone invariance property and the decay property, it is possible to ascertain the existence of an inertial manifold for (2.1).
Theorem 3.3
Assume that \(F:H\rightarrow H\) is globally Lipschitz continuous and \(\Vert A^{-\frac{1}{2}}F(u)\Vert \le C\), \(C>0\). If the strong squeezing property holds for (2.1), then there exists an N-dimensional inertial manifold \(\mathcal {M}\) as in Definition 2.3 which is the graph of a Lipschitz continuous function \(\Phi \) from \(P_{N}H\) to \( Q_{N}H\).
Proof
Inspired by the method in [44], we have four steps here to achieve our result.
\(\mathbf {Step\ 1.}\) In this step we prove that \(\forall T>0, u_{0}\in P_{N}H\), there exists a unique backward solution for the following problem on \((-\infty ,0]\):
Denote by S(t) the semigroup generated by the abstract equation (2.1) and define \(G_{T}:P_{N}H\rightarrow P_{N}H\) by
Then we need to show that \(G_{T}\) is Lipschitz continuous.
Set \(\phi _{1}, \phi _{2}\in P_{N}H\) and \(u_{1}(T)=S(T)\phi _{1}\), \(u_{2}(T)=S(T)\phi _{2}\). Then by (3.17), we get
Applying the uniform Gronwall inequality, we have
so that it follows that
which shows that \( G_{T}\) is globally Lipschitz continuous.
Next, we need to verify that \( G_{T}\) is an injective mapping. Set \(w(t)=u_{1}(t)-u_{2}(t)\), \(p=P_{N}w\), \(q=Q_{N}w\). From (3.18), we see that
and taking the inner product of (3.19) with p, we get
Since \(\Vert F(u_{1})-F(u_{2})\Vert \le L\Vert u_{1}-u_{2}\Vert =L\Vert w(t)\Vert \), one has
Since \(w(0)=u_{1}(0)-u_{2}(0)\in P_{N}H\), and application of the cone invariance property implies that \(\forall t\ge 0\), \(\Vert q\Vert ^{2}\le \Vert p\Vert ^{2}\), and we have \(\Vert w\Vert ^{2}\le 2\Vert p\Vert ^{2}\), yielding
i.e.
By virtue of the uniform Gronwall inequality, we have
and it holds
so that \(G_{T}\) is an injective mapping and \(G^{-1}_{T}\) is Lipschitz continuous. Moreover, the open set \(G_{T}(P_{N}H)\) has no boundary.
As \(P_{N}H\) is finite dimensional, \(G_{T}\) is an injective and Lipschitz continuous mapping, and from the Brouwer degree theorem, we conclude that \(G_{T}\) is a homeomorphism from \(P_{N}H\) to \(G_{T}(P_{N}H)\). There is no boundary on \(G_{T}(P_{N}H)\) and therefore \(P_{N}H=G_{T}(P_{N}H)\). As a result, \(\forall u_{0}(t)\in P_{N}H\), \( u_{0}=G_{T}(\phi )\) is uniquely solvable, which implies that \(u(t)=S(t+T)G^{-1}_{T}(u_{0})\) is the unique solution of (3.18).
\(\mathbf {Step\ 2.}\) Let \(u_{0}\in P_{N}H\) and \(u_{T}(t)\) be the solution of (3.18), so that \(Q_{N}u_{T_{1}}(-T_{1})=Q_{N}u_{T_{2}}(-T_{2})=0\). In this step we prove the existence of the limit \({\hat{u}}(t)=\lim \limits _{T\rightarrow \infty }u_{T}(t)\) which is a backward solution of the following equation:
Set \(T_{2}>T_{1}>0\), \(w(t)=u_{T_{1}}(t)-u_{T_{2}}(t)\), and \(\forall t\in [-T_{1},0]\), \(p=P_{N}w, q=Q_{N}w\). Clearly, \(p(0)=P_{N}u_{T_{1}}(0)-P_{N}u_{T_{2}}(0)=0\). Due to the cone invariance property, \(\forall t\in [-T_{1},0]\), \(\Vert q\Vert \ge \Vert p\Vert \), so that \(2\Vert q\Vert \ge \Vert w\Vert \). When \(\Vert q\Vert \ge \Vert p\Vert \), by the decay property and since \(Q_{N}u_{T_{1}}(-T_{1})=0\), we see that \(\forall t\in [-T_{1},0],\)
Next, we show that \(\forall T_{2}>T_{1}>0\), \(\Vert Q_{N}u_{T_{2}}(-T_{1})\Vert \) is uniformly bounded. For any solution \(u_{T}\) of (3.18), taking the inner product of (3.20) with \(Q_{N}u_{T}\) and using the Cauchy inequality with \(\epsilon \), we get
In view of (3.21), this yields
When \(\exists C>0\) such that \(\Vert A^{-\frac{1}{2}}F(u_{T})\Vert \le C\), it follows that \(\forall t \in [-T, 0],\)
By the uniform Gronwall inequality and since \(Q_{N}u_{T}(-T)=0\), we conclude that \(\forall t \in [-T, 0]\), \(T>0\), \(\Vert Q_{N}u_{T}(t)\Vert ^{2}\le C\).
This implies that \(\forall t \in [-T_{1}, 0]\), \(T_{2}>T_{1}>0\), one has
Let \(T_{1}\rightarrow +\infty \), so that \(\Vert u_{T_{1}}(t)-u_{T_{2}}(t)\Vert \rightarrow 0\) and \(u_{T}\) is a Cauchy sequence in \(\mathcal {C}([-T_{1}, 0]; H)\). Therefore there exists \({\hat{u}}\) making \({\hat{u}}(t)=\lim \limits _{T\rightarrow \infty }u_{T}(t)\) and we can conclude that \({\hat{u}}\) is the backward solution of (3.20) with a unique extension \({\hat{u}}(t)\) solving (2.1) for \(t\in \mathbb {R}\), and \(\Vert Q_{N}{\hat{u}}(t)\Vert ^{2}\le C\), \(\forall t\in \mathbb {R}\).
\(\mathbf {Step\ 3.}\) In this step we construct a Lipschitz continuous function \(\Phi :P_{N}H\rightarrow Q_{N}H\), and \(\mathcal {M}:=Graph(\Phi )=\left\{ {p+\Phi (p),p\in P_{N}H}\right\} \).
Define \(\mathcal {A}\) as the set of all \({\hat{u}}(t), t\in \mathbb {R}\). From the properties of \({\hat{u}}(t)\) in the above step, we know that \({\hat{u}}(t)\) is the solution of (2.1), and \(\mathcal {A}\) is invariant, \(S(t)\mathcal {A}=\mathcal {A},\forall t\ge 0\). By the construction of \({\hat{u}}(t)\), there exist \(u_{0}^{1}, u_{0}^{2}\in P_{N}H\) such that \({\hat{u}}_{1}(t)=\lim \limits _{T\rightarrow \infty }u_{T, u_{0}^{1}}(t)\), \({\hat{u}}_{2}(t)=\lim \limits _{T\rightarrow \infty }u_{T, u_{0}^{2}}(t)\), where \(u_{0}^{1}=P_{N}u_{T, u_{0}^{1}}(0)\), \(u_{0}^{2}=P_{N}u_{T, u_{0}^{2}}(0)\), and we still have \(Q_{N}u_{T, u_{0}^{1}}(-T)=Q_{N}u_{T, u_{0}^{2}}(-T)=0\), together with the cone invariance property. We claim that \(\forall t\ge -T,\)
Let \(T\rightarrow \infty \). Then
Define \(\Phi :P_{N}H\rightarrow Q_{N}H\), so that \(\Phi (u_0)=Q_{N}{\hat{u}}(0)\), where \({\hat{u}}\in \mathcal {A}\), \(u_0=P_{N}{\hat{u}}(0)\). We obtain
Obviously, \(\Phi \) is Lipschitz continuous, with Lipschitz constant 1.
Therefore, we can construct the \(N-\)dimensional Lipschitz manifold as
and the invariance of \(\mathcal {M}\) can be deduced from that of \(\mathcal {A}\).
\(\mathbf {Step\ 4.}\) Here, we shall prove that any solution of (2.1) will approach the \(N-\)dimensional Lipschitz manifold \(\mathcal {M}\) exponentially fast.
Suppose that \(u(t), t \ge 0\) is the solution of (2.1), and \(u(t)\notin \mathcal {M}\). \(\forall T \>0\). There exists \({\hat{u}}_{T}\in \mathcal {A}\) such that \(P_{N}u(T)=P_{N}{\hat{u}}_{T}\). By the cone invariance property and the decay property, we have \(\forall t\in [0, T],\)
and
As a result of Step 2, \(\forall T>0\), \(\Vert {\hat{u}}_{T}(0)\Vert \) is uniformly bounded. Since \(\mathcal {M}\) is finite dimensional, and \({\hat{u}}_{T}(0)\in \mathcal {M}\), one has a sequence \({\hat{u}}_{T_{j}}(0)\rightarrow {\hat{u}}(0)\) as \(T_{j}\rightarrow \infty \). Furthermore, the corresponding trajectory \({\hat{u}}_{T_{j}}(t)\in \mathcal {A}\). For \({\hat{u}}_{T_{j}}, {\hat{u}}\) solutions of (2.1), owing to (3.17), it follows that
When \(T_{j}\rightarrow \infty \), we have \(\Vert {\hat{u}}_{T_{j}}(t)-{\hat{u}}(t)\Vert \rightarrow 0\). With the help of (3.22), we obtain
This means that any solution u(t) out of \(\mathcal {M}\) can approach an orbit \({\hat{u}}(t)\) that belongs to \(\mathcal {M}\). The proof is complete. \(\square \)
Thus, by the above result, we conclude that if the spatial averaging condition of Theorem 3.1 holds, then there exists an inertial manifold for the model problem (2.1) with common conditions on A and F.
4 Abstract Invariant Manifolds Theorem for a PDE-ODE Coupled System
In this section, the abstract invariant manifold theorem, as presented in [37], will be employed to address an abstract differential system associated with the coupled equations. The spatial averaging principle was previously applied to an abstract PDE in Sect. 3, where conditions ensuring the existence of Lipschitz invariant manifolds for the PDE were identified. The emphasis in this discussion will be on the conditions for the entire system, with particular attention to those associated with the ODE.
4.1 Abstract Differential System
Let H be a Hilbert space and \(L^{\infty }(\Omega )\) be the space of functions with essential bound, which is a Banach space; their norms are \(\Vert \cdot \Vert \), \(\Vert \cdot \Vert _{L^{\infty }(\Omega )}\) respectively, and the inner product of H is \((\cdot ,\cdot )\). Define the norm on the product space \(H\times L^{\infty }(\Omega )\) by \(\Vert \cdot \Vert ^2_{H\times L^{\infty }(\Omega )}=\Vert \cdot \Vert ^2+\Vert \cdot \Vert ^2_{L^{\infty }(\Omega )}\). Here we have \(p\in P_{N}H\), \(q=Q_{N}H\) and note that \(\mathcal {P}=P_{N}H\), \(\mathcal {Q}=Q_{N}H\), \(\forall u=(p, q)\in H\), \(w, w', v\in L^{\infty }(\Omega )\). In view of system (1.2), we can consider the following abstract differential system:
Given that the solution to (1.1) is represented as \((u, v)=(p, q, v)\), the addition of an additional operator into the supporting equation is unnecessary. Let (p, q, v, w) denote the solution to (4.1).
Assume that A is the positive definite operator defined on \(\mathcal {Q}\) with compact resolvent set. It is a closed operator and the domain \(D(A)\in \mathcal {Q}\) is dense in \(\mathcal {Q}\), so that \(-A\) can generate an analytic semigroup \(e^{-At}\), \(F:H\times L^{\infty }(\Omega ) \rightarrow \mathcal {P}\), \(G:H\times L^{\infty }(\Omega ) \rightarrow \mathcal {Q}\), \(\mathcal {H}:\mathcal {P} \times \mathcal {Q} \rightarrow H\). Suppose that F, G, \(\mathcal {H}\) are bounded and Lipschitz continuous and B(w) is positive and uniformly bounded in \(L^{\infty }(\Omega )\),
and is Lipschitz continuous,
Note that \(\mathcal {H}(p, q)\) is also positive and uniformly bounded in H,
here \(a_{1}, a_{2}>0\).
From the above discussion, we can conclude that there exists a unique solution for given initial value \((p(0), q(0), v(0), w(0))=(p_{0}, q_{0}, v_{0}, w_{0})\in \mathcal {P} \times \mathcal {Q} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega ).\)
Theorem 4.1
There exists a unique mild solution for \(t \in [0, \infty )\) to the initial problem associated with (4.1),
and
where \(T\>0\). Furthermore, we can define the semigroup S(t) on \(\mathcal {P} \times \mathcal {Q} \times L^{\infty }(\Omega )\times L^{\infty }(\Omega )\) as
The proof of Theorem 4.1 relies on standard semigroups techniques, see [18, 35].
4.2 Some Important Assumptions
In this section, we give assumptions ensuring the existence of a Lipschitz invariant manifold \(\mathcal {M}\), for the abstract PDE, expressed as:
where \(\Psi \) is Lipschitz continuous.
Given the previously established existence of an inertial manifold \(\mathcal {M}\), for the abstract PDE, \(\mathcal {M}=\text{ Graph }(\Phi )\), \(\Phi : P_{N}H \rightarrow Q_{N}H\), where \(\Phi \) is Lipschitz continuous, we can deduce that there must be another Lipschitz continuous function \(\phi :L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\rightarrow \mathcal {Q}\) which satisfies
Consequently, the task will be to verify the existence of \(\Psi \) through the application of an abstract invariant manifold theorem, particularly focusing on the conditions related to the ODE. It should be noted that conditions associated with the PDE were previously addressed in Sect. 3.
Here and below, \(\mathcal {L}(X, Y)\) is the Banach space of bounded linear operators from X to Y, \({\overline{S}}\) is the closure of \(S\subset X\) with an appropriate norm, and X, Y are Banach spaces with norms \(\Vert \cdot \Vert _{X}, \Vert \cdot \Vert _{Y}\), respectively.
We first recall the definition of the weak Gateaux derivative.
Definition 4.1
[39] Let X, Y, Z be Banach spaces, \(\mathcal {K}\subset X\) be a dense subspace, and \(\mathcal {R}\subset Z\). We assume that \(\mathcal {K}\) with norm \(\Vert \cdot \Vert _{\mathcal {K}}\) is a Banach space which is continuously embedded into X. A function \(F: X\times Z \rightarrow Y\) is said to be weakly Gateaux differentiable on \(\mathcal {K}\times \mathcal {R}\) with respect to \(x\in \mathcal {K}\) if the following two conditions are satisfied:
(1) for any \(h\in \mathcal {K}\), one has \(\lim \limits _{t\rightarrow 0}\frac{F(x+th, z)-F(x, z)}{t}=L(x, z)h\), as \(t\rightarrow 0\);
(2) the linear operator L(x, z) is continuous on \(\mathcal {K}\times \mathcal {R}\) with respect to \(x\in \mathcal {K}\) in the following sense: let \(R >0\) be any fixed constant; for any \(\epsilon >0\), there is an \(\eta \>0\) such that for any \(x_{1}, x_{2}\in \mathcal {K}\) with \(\Vert x_{1}\Vert _{\mathcal {K}}, \Vert x_{2}\Vert _{\mathcal {K}} \le R\), \(z\in \mathcal {R}\), we have
whenever \(\Vert x_{1}-x_{2}\Vert _{X}\le \eta \).
We call L(x, z) the weak Gateaux derivative of F with respect to x at (x, z), and let DF(x, z) denote L(x, z).
Then we place some restrictions on \(\Vert p \Vert \), \(\Vert Aq \Vert \), \(\Vert v \Vert _{L^{\infty }(\Omega )}, \Vert w \Vert _{L^{\infty }(\Omega )}\). Namely, \(\exists R_{1}, R_{2}>0\) such that
We make the following important assumptions:
(1) (Regularity condition) \(\forall (p, q)\in (\mathcal {P}\times D(A))\), let F, G be weakly Gateaux differentiable with respect to (p, q).
(2) (Dissipativity condition) Suppose that \(p \in \overline{\mathcal {P}{\setminus } \mathcal {A}}\). Then \(\forall v\in L^{\infty }(\Omega )\),
and if \(v \in \overline{L^{\infty }(\Omega )\setminus \mathcal {C}}\), with \(\forall (p, q)\in H=\mathcal {P}\oplus \mathcal {Q}\), and \(\forall w, v\in L^{\infty }(\Omega )\), then
(3) (Sobolev condition) Let M be a constant. Then
\(\forall \big (p(t), v(t), w(t)\big )\in C\big ([0, \infty ); \mathcal {A}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\big )\cap C^{1}\big ([0, \infty ); \mathcal {A}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\big )\), and \(\forall t_{0} >0\), we have
so that the solution q(t) of the following initial problem lies in \(\mathcal {B}\) when \(t\in [0, t_{0}]\):
(4) (Uniform cone condition) Define the Lyapunov function \(V=\frac{1}{2}\Vert \sigma \Vert ^{2}-\frac{1}{2}\Vert \rho \Vert ^{2}\), where \((\rho ,\sigma )\in \mathcal {P}\times \mathcal {Q}\), \(\exists \delta >0\), \(\forall \ (p, q, v)\in \mathcal {A}\times \mathcal {B}\times \mathcal {C}\). We have
where \(\sigma ', \rho '\) are as follows:
DF(p, q, v), DG(p, q, v) being the weak Gateaux derivatives of F(p, q, v), G(p, q, v) with respect to (p, q) on \((\mathcal {P}\times D(A))\).
Remark 4.1
Since we have already proved the uniform cone condition for the abstract PDE in Sect. 3,
and the decay property
there must be a \(\delta \) corresponding to (4.3), and it has no relation to the spectral gap condition \((\lambda _{N+1}-\lambda _{N} >2L)\).
(5) (Linear stable condition) Let \((q(t), Aq(t)) \ge \lambda _{N+1}\Vert q(t)\Vert ^{2}\) and assume that \(\forall (p, q, v) \in \mathcal {P}\times \mathcal {Q}\times L^{\infty }(\Omega ) \)
here L is a constant which satisfies that \(\exists N \in \mathbb {N}\) such that
These assumptions are inspired from [31]. The subsequent sections present essential definitions and representations prerequisite for the proof of the abstract invariant manifold theorem.
Define \(\Psi :\mathcal {P}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega ) \rightarrow \mathcal {Q}\). The graph and support of \(\Psi \) are as follows:
Denote some subsets of \(H\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\) by
where \(\partial \mathcal {A}, \partial \mathcal {C}, \partial \mathcal {D}\) are the boundaries of \(\mathcal {A}\), \(\mathcal {C}\), \(\mathcal {D}\), respectively.
4.3 Invariant Manifold for the Abstract Differential System
Theorem 4.2
Suppose that (4.1) satisfies the five conditions above. Then \(\exists \Psi :\mathcal {P} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\rightarrow \mathcal {Q}\), \(\forall (p, v, w)\in \mathcal {P} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega ) \)
such that \(\mathcal {M}=Graph(\Psi )\subset \mathcal {K}\) is the Lipschitz invariant manifold of (4.1). Furthermore, the manifold \(\mathcal {M}\) is locally attracting in the following sense: \(\exists \alpha , C >0\) such that \(\forall t\ge 0\), setting \(y(t)=(p(t), q(t), v(t), w(t))\in \mathcal {G}\), we have
The proof of this theorem mainly comes from the Hadamard graph transform method. We start with the flat manifold \(\mathcal {M}_{0}=\left\{ \mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega ) \right\} \), and let the dynamics of (4.1) act on \(\mathcal {M}_{0}\), so that we can get \(\mathcal {M}_{t}\) as the image of \(\mathcal {M}_{0}\) under the semiflow of (4.1), \(\forall t >0\).
Here \(\mathcal {M}_{t}=\text{ Graph }(\Psi _{t})\), \(\Psi _{t}: \mathcal {P} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega ) \rightarrow \mathcal {Q}\), and we have \(\lim \limits _{t\rightarrow +\infty }\Psi _{t}=\Psi \). Finally we will show that the desired manifold is \(\mathcal {M}=\text{ Graph }(\Psi )\).
Note that
so that \(q_{t}=\Psi _{t}(p_{t}, v_{t}, w_{t})\) and
Next we will prove Theorem 4.2 through some lemmas, and we first find out the space in which the solution y(t) lies when given an initial value y(0).
Lemma 4.1
If \(y(0)=(p_{0}, 0, v_{0}, w_{0})\in \mathcal {A}\times \left\{ 0\right\} \times \mathcal {C} \times \mathcal {D}\), then \(\forall t\in [0, \infty )\)
Proof
This proof mainly consists of three steps, and each step has different situations to consider.
\(\mathbf { Step\ 1 }:\) We first prove that \(\forall t>0\),
Due to \(\Vert q_{0}\Vert =0\), and \(dist(p,\ \partial \mathcal {A})=0\), if \(p\in \partial \mathcal {A}\), we have the following two cases to discuss.
\(\mathbf {Case\ a}\): \(t>0\), but t is small.
If \(p_{0}\in int(\mathcal {A})\), interior of A, we have
The above inequality holds obviously for \(\Vert q_{0}\Vert =0\) and t small.
If \(p_{0}\in \partial \mathcal {A}\), by virtue of the dissipativity condition, we have
This implies that \(p_{t}\) leaves \(\partial \mathcal {A}\) when \(t=0\). Moreover
and we conclude that (4.4) holds in this case.
\(\mathbf {Case\ b}\): Let \(t_{0}\) be the largest time that makes (4.4) hold for \(\forall t\in [0, t_{0}]\). We will show that \(t_{0}\rightarrow \infty \).
Indeed, we shall prove Case b by contradiction. Suppose that \(t_{0}<\infty \), and we have
Here we have two situations to consider, \(\Vert q_{t_{0}}\Vert \ne 0\), or \(p_{t_{0}}\in \partial \mathcal {A}\). However, from Case a we know that when \(p_{t_{0}}\in \partial \mathcal {A}\), (4.4) holds for \(t>t_{0}\) near \(t_{0}\), which contradicts our assumption on \(t_{0}\). Thus \(p_{t_{0}}\in \partial \mathcal {A}\). Set \(p_{1}\in \partial \mathcal {A}\), and
Hence \(\exists \gamma \in [0, 1)\), \(p_{t_{0}}=\gamma p_{1}\); actually \(\gamma =\frac{\Vert p_{t_{0}}\Vert }{R_{1}}\).
Define \(\rho (t)=p_{t}-p_{1}\), \(\sigma (t)=q_{t}-0=q_{t}\). From the regularity and dissipativity conditions, we know that
and \(V(t)=\frac{1}{2}\Vert \sigma \Vert ^{2}-\frac{1}{2}\Vert \rho \Vert ^{2}\), so that \(V(t_{0})=0\). In addition, according to the assumptions, we have
By the dissipativity and uniform cone conditions, together with \(p_{t_{0}}=\gamma p_{1}\), when \(t=t_{0}\) we can deduce that
This implies that V(t) decreases strictly when \(t=t_{0}\), which contradicts (4.6). Consequently, \(t_{0}=\infty \), and we arrive at the conclusion that (4.4) is valid.
\(\mathbf { Step\ 2 }:\) Next we prove that \(\forall t\ge 0\)
As in Step 1, we also have two cases to consider in this situation, and we shall show that \(t\rightarrow \infty \).
\(\mathbf {Case\ \uppercase {i}}\): \(t>0\) but t is small.
If \(v_{0}\in int(\mathcal {C})\), we have
for \(\Vert v_{0}\Vert =0\), and (4.7) is valid when t is small.
If \(v_{0}\in int(\partial {C})\), by the dissipativity condition,
which means that
so that \(v_{0}\) is going away from \(\partial \mathcal {C}\).
On the other hand,
and we can see that (4.7) holds in Case I.
\(\mathbf {Case\ \uppercase {ii}}\): Set \(t_{1}\) as the largest time making (4.7) hold for \(\forall t\in [0, t_{1}]\); we also have \(t_{1}\rightarrow \infty \).
Suppose that \(t_{1}<\infty \). Then
and we must have \(\Vert q(t_{1})\Vert \ne 0\), or \(v(t_{1})\in \partial \mathcal {C}\). However from Case I we know that (4.7) holds when \(t>t_{1}\) and near \(t_{1}\), which contradicts our setting on \(t_{1}\), so that \(v(t_{1})\in int (\mathcal {C})\).
Let \(v_{1}\in \partial \mathcal {C}\), and
Clearly we can deduce that \(\gamma _{1}\in [0, 1)\), so that \(v(t_{1})=\gamma _{1}v_{1}\); actually we have \(\gamma _{1}=\frac{\Vert v(t_{1})\Vert _{L^{\infty }(\Omega )}}{R_{1}}\).
Define \(\rho (t)=v(t)-v_{1}\), \(\sigma =q(t)-0=q(t)\), so that
Define \(V(t)=\frac{1}{2}\Vert \sigma \Vert ^{2}-\frac{1}{2}\Vert \rho \Vert ^{2}_{L^{\infty }(\Omega )}\). Thus \(V(t_{1})=0\), and \(\forall t \in [0, t_{1}]\), \(V(t)<0\). By the dissipativity and linear stability conditions, together with \(v(t_{1})=\gamma _{1}v_{1}\), when \(t=t_{1}\), we have
This implies that V(t) is strictly decreasing at \(t_{1}\), contradicting the assumption in Case II that \(t_{1}<\infty \).
Consequently, \(t_{1}\rightarrow \infty \), and (4.7) is also valid in Case II.
\(\mathbf { Step\ 3 }:\) Finally, we prove that \(\forall t\ge 0\),
As above, we have two cases to consider as well, and \(t\rightarrow \infty \).
\(\mathbf {Case\ A}\): Let \(t>0\) but be small.
If \(w_{0}\in int(\mathcal {D})\), we have
for \(\Vert w_{0}\Vert =0\), and (4.7) is valid when t is small.
When \(w_{0}\in int(\partial {D})\), from the dissipativity condition,
which means that
so that \(w_{0}\) goes away from \(\partial \mathcal {D}\), together with
and (4.7) is valid in this case.
\(\mathbf {Case\ B}\): Let \(t_{w}\) be the largest time making (4.7) hold \(\forall t\in [0, t_{w}]\), and \(t_{w}\) goes to infinity.
Suppose that \(t_{w}<\infty \). Then
so that \(\Vert q(t_{1})\Vert \ne 0\), or \(w(t_{w})\in \partial \mathcal {D}\). In addition, from Case A we know that (4.7) holds when \(t>t_{w}\) and near \(t_{w}\), which contradicts the definition of \(t_{w}\), so that \(w(t_{w})\in int (\mathcal {D})\).
Let \(w_{1}\in \partial \mathcal {D}\), and
We can see that \(\gamma _{2}\in [0, 1)\), so that \(w(t_{w})=\gamma _{2}w_{1}\); in fact, we have \(\gamma _{2}=\frac{\Vert w(t_{w})\Vert _{L^{\infty }(\Omega )}}{R_{1}}\).
Define \(\rho (t)=w(t)-w_{1}\), \(\sigma =q(t)-0=q(t)\), so that
Define \(V(t)=\frac{1}{2}\Vert \sigma \Vert ^{2}-\frac{1}{2}\Vert \rho \Vert ^{2}_{L^{\infty }(\Omega )}\). Thus \(V(t_{w})=0\), and \(\forall t \in [0, t_{w}]\), \(V(t)<0\). By the dissipativity and linear stability conditions, together with \(w(t_{w})=\gamma _{2}w_{1}\), when \(t=t_{w}\), we have
This implies that V(t) is strictly decreasing at \(t_{w}\), contradicting the assumption in Case II that \(t_{w}<\infty \).
Consequently, \(t_{w}\rightarrow \infty \), (4.7) is also valid in Case B. This completes the proof. \(\square \)
Next, we expand the space in which y(0) defined in Lemma 4.1 lies, from \(\mathcal {A}\times \left\{ 0\right\} \times \mathcal {C} \times \mathcal {D}\) to \(\mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\), and show that \(y(t)\in \mathcal {K}\). Moreover, \(\mathcal {M}_{t}\in \mathcal {K}\).
Lemma 4.2
If \(y(0)=(p_{0}, 0, v_{0}, w_{0})\in \mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\), then \(\forall t\in [0, \infty )\),
Proof
From Lemma 4.1, we know that if \((p_{0}, 0, v_{0}, w_{0})\in \mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\), then the solution of (4.1) \(y(t)\in \mathcal {G}\), \(\forall t \ge 0\). In addition, the dissipativity condition illustrates that when \((p_{0}, v_{0}, w_{0})\in (\mathcal {P}{\setminus } \mathcal {A})\times (L^{\infty }(\Omega ){\setminus } \mathcal {C})\times (L^{\infty }(\Omega ){\setminus }\mathcal {D})\), \(t>0\), and near 0, \((p_{t}, v_{t}, w_{t})\) will approach \(\mathcal {A}\times \mathcal {C}\times \mathcal {D}\). Set \(R_{1}\) large enough, so that we have \(y(t)\in \mathcal {G}\). By the definition of \(\mathcal {M}_{t}\), we can deduce that \(\mathcal {M}_{t}\subset \mathcal {G}\) when \(t>0\), \(\mathcal {M}_{t}\subset \mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\) when \(t=0\). We can obtain Lemma 4.2 from the above discussion, which completes the proof. \(\square \)
Through Lemmas 4.1 and 4.2 we know that y(t) exists in \(\mathcal {K}\) when \(y(0)\in \mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\). In the next lemma, we prove that any y(t) enters the smaller space \(\mathcal {G}\) after a certain time \(t_{0}\).
Lemma 4.3
Suppose that \(y(0)=(p_{0}, 0, v_{0}, w_{0})\in \mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\) and denote the solution of (4.1) by \(y(t)=(p_{t}, q_{t}, v_{t}, w_{t})\). There exists \( t_{0}>0\) such that
(1) at least one solution y(t) is outside \(\mathcal {A}\times \mathcal {B} \times \mathcal {C} \times \mathcal {D}\) when \(t\in [0, t_{0})\);
(2) all \(y(t)\in \mathcal {G}\) if \(t\ge t_{0}\).
Proof
For all \(y(0)\in \mathcal {P}\times \left\{ 0\right\} \times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\), we have \(q(0)=0\). By Lemma 4.2 and the dissipativity condition, if \((p_{0}, v_{0}, w_{0})\in (\mathcal {P}{\setminus } \mathcal {A})\times (L^{\infty }(\Omega ){\setminus } \mathcal {C})\times (L^{\infty }(\Omega ){\setminus } \mathcal {D})\), \((p_{t}, v_{t}, w_{t})\) will get close to \(\mathcal {A} \times \mathcal {C} \times \mathcal {D}\); here we have \(y(t)\notin \mathcal {A}\times \mathcal {B} \times \mathcal {C} \times \mathcal {D}\). Until \(t=t_{0}\), \((p_{t}, v_{t}, w_{t})\) enters \(\mathcal {A}\times \mathcal {C}\times \mathcal {D}\). Thus, together with Lemma 4.1, (2) is obviously valid and the proof is complete. \(\square \)
Combining the three lemmas above, we know that \(\mathcal {M}_{t}\subset \mathcal {G}\), and all solutions y(t) enter a bounded set after a certain time, which means that \(\Vert p(t)\Vert , \Vert q(t)\Vert , \Vert v(t)\Vert , \Vert w(t)\Vert \) are all bounded after \(t_{0}\).
Set \(y_{1}(t), y_{2}(t)\) to be any two solutions of (4.1). Then we consider the relations between \(\Vert q_{1}(t)- q_{2}(t)\Vert \), \(\Vert p_{1}(t)- p_{2}(t)\Vert \), \(\Vert v_{1}(t)- v_{2}(t)\Vert _{L^{\infty }(\Omega )}\), and \(\Vert w_{1}(t)- w_{2}(t)\Vert _{L^{\infty }(\Omega )}\), which is a preliminary step to construct the desired manifold \(\mathcal {M}_{t}\).
As in the proof of the strong squeezing property theorem, there also are two situations that need to be discussed,
and
Next we start with (4.8).
Lemma 4.4
Let \(y_{1}(t), y_{2}(t)\in \mathcal {K}\) be two solutions of (4.1). If there exists \(t_{1}\ge 0\) such that (4.8) holds, then (4.8) is valid for \(\forall t\ge t_{1}\).
Proof
Indeed, this lemma is analogous to the cone invariance property, but due to \(q_{1}(0)=q_{2}(0)=0\), we cannot calculate the inequalities when \(t=0\). We thus set \(t_{1}\ge 0\) for the explicit expression, which is selected freely.
We have the following two cases.
\(\mathbf {Case\ 1}\): If \(\max \{\Vert v_{1}(t_{1})- v_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}, \Vert w_{1}(t_{1})- w_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}\}\le \Vert p_{1}(t_{1})- p_{2}(t_{1})\Vert \), then
and by the uniform cone condition, together with the cone invariance property, we can see that
holds for \(t\ge t_{1}\).
\(\mathbf {Case\ 2}\): If \(\Vert p_{1}(t_{1})- p_{2}(t_{1})\Vert \le \max \{\Vert v_{1}(t_{1})- v_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}, \Vert w_{1}(t_{1})- w_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}\}\) and, moreover, \(\Vert w_{1}(t_{1})- w_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}\le \Vert v_{1}(t_{1})- v_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}\), we have
since \(t_{1}\) can be freely selected. By Lemma 4.3, we just need to prove that
holds for \(t\in [0, t_{0})\) and \([t_{0}, \infty )\).
From the definition of \(t_{0}\), we know that it is the first time for \(\mathcal {G}\) to contain all solutions, so that \(y_{1}(t), y_{2}(t)\) may enter \(\mathcal {G}\) at different times.
Without loss of generality, assume that \(y_{2}(t)\) goes into \(\mathcal {G}\) at \(t_{2}<t_{0}\). Thus we have
\(\mathbf {Case\ 2-1}\): If \(t\in [0, t_{2})\), then the result holds obviously. If \(t\in [t_{2}, t_{0})\), we have \(q_{1}(t)= 0\), and \(y_{2}(t)\in \mathcal {G}\), \(v_{2}(t)\in \mathcal {C}\), \(y_{1}(t)\notin \mathcal {G}\), so that \(\Vert q(t)\Vert \le dist(v(t), \partial \mathcal {C})\).
When \(v_{1}(t)\notin \mathcal {C}\), we get
namely, \(\Vert q_{1}(t)- q_{2}(t)\Vert \le \Vert v_{1}(t)- v_{2}(t)\Vert _{L^{\infty }(\Omega )}.\)
If \(p_{1}(t)\notin \mathcal {A}\), in the same way we have \(\Vert q_{1}(t)- q_{2}(t)\Vert \le \Vert p_{1}(t)- p_{2}(t)\Vert \), and we go back to the discussion in Case 1.
\(\mathbf {Case\ 2-2}\): Assume that (4.10) is not valid for \(t\in [t_{0}, \infty )\).
Define \(\rho (t)=v_{1}(t)- v_{2}(t)\), \(\sigma (t)=q_{1}(t)- q_{2}(t)\). We now have the following two precise cases to discuss.
\(\mathbf {Case\ 2-2-1}\): If there exists \(t_{3}\ge t_{0}\) such that
and when \(t=t_{3}\), \(\Vert q_{1}(t_{3})- q_{2}(t_{3})\Vert =\Vert v_{1}(t_{3})-v_{2}(t_{3})\Vert _{L^{\infty }(\Omega )}\), while \(t>t_{3}\), we have \(\Vert q_{1}(t)- q_{2}(t)\Vert >\Vert v_{1}(t)-v_{2}(t)\Vert _{L^{\infty }(\Omega )}\).
Since \(V(t)|_{t=t_{3}}=0\), using the linear stability condition,
which contradicts our assumption.
\(\mathbf {Case\ 2-2-2}\): If there exists \(t_{3}\ge t_{0}\) such that
we go back to Case 1 again, and the result contradicts our assumptions as well.
\(\mathbf {Case\ 3}\): If \(\Vert p_{1}(t_{1})- p_{2}(t_{1})\Vert \le max\{\Vert v_{1}(t_{1})- v_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}, \Vert w_{1}(t_{1})- w_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}\}\), on the contrary, \(\Vert v_{1}(t_{1})- v_{2}(t_{1})\Vert _{L^{\infty }(\Omega )}\le \Vert w_{1}(t)- w_{2}(t)\Vert _{L^{\infty }(\Omega )}\), so that
and the proof is similar to Case 2.
Due to the arbitrariness of \(t_{1}\) and Lemma 4.3, we shall prove that
holds for \(t\in [0, t_{0})\) and \([t_{0}, \infty )\).
Assume that \(y_{2}(t)\) goes into \(\mathcal {G}\) at \(t_{2}<t_{0}\),
\(\mathbf {Case\ 3-1}\): If \(t\in [0, t_{2})\), then \(q_{1}(t)= q_{2}(t)=0\), and (4.11) holds. If \(t\in [t_{2}, t_{0})\), we have \(q_{1}(t)= 0\), and \(y_{2}(t)\in \mathcal {G}\), \(w_{2}(t)\in \mathcal {D}\), \(y_{1}(t)\notin \mathcal {G}\), so that \(\Vert q(t)\Vert \le dist(w(t), \partial \mathcal {D})\).
When \(w_{1}(t)\notin \mathcal {C}\), we have
that is, \(\Vert q_{1}(t)- q_{2}(t)\Vert \le \Vert w_{1}(t)- w_{2}(t)\Vert _{L^{\infty }(\Omega )}.\)
The possible evidence, \(p_{1}(t)\notin \mathcal {A}\), leads to the discussion in Case 1 again.
\(\mathbf {Case\ 3-2}\): Assume that (4.11) is not valid for \(t\in [t_{0}, \infty )\).
Define \(\rho (t)=w_{1}(t)- w_{2}(t)\), \(\sigma (t)=q_{1}(t)- q_{2}(t)\). The following two cases arise.
\(\mathbf {Case\ 3-2-1}\): If there exists \(t_{3}\ge t_{0}\) such that
and when \(t=t_{3}\), \(\Vert q_{1}(t_{3})- q_{2}(t_{3})\Vert =\Vert w_{1}(t_{3})-w_{2}(t_{3})\Vert _{L^{\infty }(\Omega )}\), while \(t>t_{3}\), we have \(\Vert q_{1}(t)- q_{2}(t)\Vert >\Vert w_{1}(t)-w_{2}(t)\Vert _{L^{\infty }(\Omega )}\).
Since \(V(t)|_{t=t_{3}}=0\), using the linear stability condition,
which leads to a contradiction.
\(\mathbf {Case\ 3-2-2}\): If \(\exists \ t_{3}\ge t_{0}\) such that
this falls into Case 1, and the result contradicts our assumption as well.
In conclusion, Lemma 4.4 holds and the proof is complete. \(\square \)
The last lemma demonstrates that we can have an important property for (4.1), which is similar to the cone invariance property. In other words, if there exists \(t_{x}\in (t_{1}, \infty )\) such that \(V(t_{x})>0\), we must have \(V(t)>0\) for any \(t\in (0, t_{x})\), or we have \(V(t)<0\) for \(t\in (t_{x}, \infty )\) if \(V(t_{x})<0\).
Furthermore, by Lemma 4.4 and the strong squeezing property theorem, we claim that we always have \(V'(t)<0\) whenever \(V(t)=\Vert q_{1}(t)- q_{2}(t)\Vert - \Vert v_{1}(t)- v_{2}(t)\Vert _{L^{\infty }(\Omega )}\) or \(V(t)=\Vert q_{1}(t)- q_{2}(t)\Vert - \Vert p_{1}(t)- p_{2}(t)\Vert \), or \(V(t)=\Vert q_{1}(t)- q_{2}(t)\Vert - \Vert w_{1}(t)- w_{2}(t)\Vert _{L^{\infty }(\Omega )}\). Despite we might have \(V(t)>0\), it will decrease to 0 exponentially fast. So next we just need to show that for y(t) and \(\Vert q_{1}(t)- q_{2}(t)\Vert \), there also have properties resembling the decay property in the strong squeezing property theorem. For \(p_{1}(t)= p_{2}(t)=0\) when \(t=0\), as is done in the above discussions, we set \(t_{5}\in (0, t_{4})\), which can be selected freely, so that the properties can be naturally extended from \((t_{5}, t_{4})\) to \((0, t_{4})\).
Lemma 4.5
Let \(y_{1}(t),y_{2}(t)\in \mathcal {K}\) be any two solutions. If \(\forall t \in (t_{5}, t_{4})\),
we have
and
Proof
Define \(\rho (t)=p_{1}(t)- p_{2}(t)\), \(\sigma (t)=q_{1}(t)- q_{2}(t)\), \(\theta (t)=v_{1}(t)- v_{2}(t)\), \(\eta (t)=w_{1}(t)- w_{2}(t)\), so that we can deduce that \(\Vert y_{1}(t)- y_{2}(t)\Vert ^2_{H \times L^{\infty }(\Omega )\times L^{\infty }(\Omega )}=\Vert \rho (t)\Vert ^2+\Vert \sigma (t)\Vert ^2+\Vert \theta (t)\Vert ^2_{L^{\infty }(\Omega )}+\Vert \eta (t)\Vert ^2_{L^{\infty }(\Omega )}\). Furthermore, from the above calculations, we have
and
Then we obtain
In addition,
and
Combining (4.12), (4.13) with (4.14) leads to
An application of the uniform Gronwall inequality gives
The proof is complete. \(\square \)
Based on the above results, we know that there must be a Lipschitz continuous function \(\Psi _t:\mathcal {P}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\rightarrow \mathcal {Q}\) such that \( \mathcal {M}_t=\text{ Graph }(\Psi _t).\) Then we just need to verify the corresponding function \(\Psi _{t}\) convergence to a function \(\Psi \), uniformly, and the corresponding manifold \(\mathcal {M}_{t}\) convergence to \(\mathcal {M}\) with the property \(\mathcal {M}=\text{ Graph } (\Psi );\) this invariant manifold is what we want.
Theorem 4.3
(Schauder’s Fixed Point Theorem) [9] Let X be a real Banach space, \(C\subset X\) be nonempty closed bounded and convex, \(F:C\longrightarrow C\) be continuous and compact. Then F has a fixed point.
Lemma 4.6
If \(y_{1}(t),y_{2}(t)\in \mathcal {K}\) are solutions satisfying Lemma 4.4, then \(\forall t\ge 0\), \(\exists \Psi _{t}:\mathcal {P}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\rightarrow \mathcal {Q}\) such that
Proof
By virtue of Lemma 4.4, we can define \(\Psi _{t}:\mathcal {P}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\rightarrow \mathcal {Q}\), that is \(\Psi _{t}(p(t), v(t),w(t))=q(t)\), which implies that \(\Psi _{t}\) satisfies the above results. From Sect. 4.2, we have
By the above analysis, we just need to prove the well-posedness of the solution, that is, for any \(y(t)=\big (p(t), q(t), v(t), w(t)\big )\in \mathcal {K}\), there exists a unique \(\big (p(0), 0, v(0), w(0)\big )\in \mathcal {P}\times L^{\infty }(\Omega )\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\) such that
Let \(p_1,v_1,w_1\) and \(t_0\) be given. We want to show that \(p_1(t_0)=p\big (t_0;(p(0), 0, v(0), w(0))\big )\), \(v_1(t_0)=v\big (t_0;(p(0), 0, v(0), w(0))\big )\), \(w_1(t_0)=w\big (t_0;(p(0), 0, v(0), w(0))\big )\) exist for some p(0), v(0), w(0) such that \(\big (p_1(t_0), q_1(t_0), v_1(t_0), w_1(t_0)\big )\in \mathcal {M}_{t_0}\) with \(q_1(t_0)=q\big (t_0;p(0), 0, v(0), w(0)\big )\).
Since the differential system (4.1) possesses the global attractor, the maps (p(t), v(t), w(t)) are continuous and compact in \(\mathcal {P}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\). Thus according to Theorem 4.3, there is (p(0), v(0), w(0)) in some ball in \(\mathcal {P}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\) with \(p_1(t_0)=p\big (t_0;(p(0), 0, v(0), w(0))\big )\), \(v_1(t_0)=v\big (t_0;(p(0), 0, v(0), w(0))\big )\), \(w_1(t_0)=w\big (t_0;(p(0), 0, v(0), w(0))\big )\) as required. The proof is complete. \(\square \)
Lemma 4.7
If \(y_{1}(t),y_{2}(t)\in \mathcal {K}\) are solutions satisfying Lemma 4.5, then \( \forall t_{5} \le t_{1} \le t_{2} \le t_{4}\), \((p, v, w)\in \mathcal {P}\times L^{\infty }(\Omega ) \times L^{\infty }(\Omega )\), \(\exists C>0\) such that
Proof
The first inequality is proved from Lemma 4.5. In addition, for \(\mathcal {B}=\{q\in \mathcal {Q}:\Vert Aq \Vert \le R_{2}\}\), together with the Poincar\(\acute{e}\) inequality and the Sobolev condition, we have \(\Vert q \Vert \le C \Vert Aq \Vert \le C R_{2}\). Then \(\Vert q(t_{1})- q(t_{2}) \Vert \le 2C R_{2}\), so that we can get the second inequality. The proof is complete. \(\square \)
Corollary 4.1
\(\{ \Psi _{t}(p(t),v(t), w(t))\}\) must be a Cauchy sequence in any cases, so that \(\lim \limits _{t\rightarrow \infty }\Psi _{t}=\Psi \) exists.
Proof
By Lemmas 4.5, 4.6, 4.7, we get the existence of \(\Psi _{t}\) which satisfies the conditions in Lemma 4.6. If \(t_{4} \rightarrow \infty \) in Lemma 4.7, then \(\{ \Psi _{t}(p(t),v(t), w(t))\}\) is a Cauchy sequence and we have \(\lim \limits _{t\rightarrow \infty }\Psi _{t}=\Psi \), while \(\Vert q(t_{1})- q(t_{2}) \Vert \) is sufficiently large here, so that it is not hard to see that \(\{ \Psi _{t}(p(t),v(t), w(t))\}\) is still a Cauchy sequence deduced from Lemma 4.4 (here \(\Vert q(t_{1})- q(t_{2}) \Vert \) is small), and \(\Psi \) also satisfies the conditions in Lemma 4.6. The proof is complete. \(\square \)
Next, we prove the invariance of \(\mathcal {M}=\text{ Graph }(\Psi )\) and the exponential tracking property.
Lemma 4.8
Suppose that \(y(t)=(p(t), q(t), v(t), w(t))\) is the solution of the system, then \(\mathcal {M}=Graph(\Psi )\) is invariant, that is, \(\exists t_{6}>0\) such that \(\Psi _{t_{6}}(p,v,w)=q(t_{6})\). Then \(\forall t\ge 0\), \(\Psi (p,v,w)=q(t)\), q is the solution of \(q'=-Aq+G(p(t), q, v(t))\), and we have \(\mathcal {M}\subset \mathcal {K}\). Meanwhile, \(\forall t>t_{5}>0\), there holds
Proof
Details on the invariance of \(\mathcal {M}\) use the same idea as in [39, Lemma 2.9].
By the invariance and Lemma 4.2, we have
Due to \(\Psi _{t}=q\), \(\Vert A\Psi _{t}\Vert \le R_{2}\), \(\lim \limits _{t\rightarrow \infty }\Psi _{t}=\Psi \), and we obtain
Then
so that we can derive that \(\mathcal {M}\subset \mathcal {K}\) holds. Combining this with Lemma 4.5, \(\forall y_{1}(t)\in \mathcal {M}\),
which leads to
The proof is complete. \(\square \)
In conclusion, based on the preceding lemmas and corollary, coupled with the five conditions stipulated in the assumptions, Theorem 4.2 is verified. This confirms the existence of an invariant manifold for the abstract differential system.
5 Application to the Coupled System
In this section, we use the results established in Sect. 4 to construct an invariant manifold for (1.1).
5.1 The Modified System
Consider the coupled system (1.1) defined on the torus \(\mathbb {T}^3\) with periodic boundary conditions. We make the following assumptions:
where \(\beta _{1},\beta _{2}, \beta _{3},C>0\). The well-posedness of the solution \((v, \partial _{t}v, u)\) in \([L^{\infty }(\mathbb {T}^3)]^{2}\times (H^{1}(\mathbb {T}^3)\cap L^{\infty }(\mathbb {T}^3))\) corresponds to Theorem 4.1. Note that these assumptions are not in contradiction with those in the abstract differential system.
Definition 5.1
A pair of functions \(\big (v(t), u(t)\big )\) is a solution of problem (1.1) if
for every \(t\ge 0\) and (1.1) is satisfied in the sense of distributions.
Theorems on the well-posedness for the weak solution and the existence of a global attractor of (1.2) are presented as follows. The comprehensive proofs can be found in [11, Chapter 11].
Theorem 5.1
[11, 12] If the initial value of (1.2) \((u_{0}, v_{0}, w_{0})\in [H^{1}(\mathbb {T}^3)\cap L ^{\infty }(\mathbb {T}^3)]\times L ^{\infty }(\mathbb {T}^3)\times L ^{\infty }(\mathbb {T}^3) \), then the problem possesses a unique weak solution
Theorem 5.2
[11, 12] The system (1.2) has the global attractor \(\mathcal {A}\in [H^{1}(\mathbb {T}^3)\cap L ^{\infty }(\mathbb {T}^3)]\times L ^{\infty }(\mathbb {T}^3)\times L ^{\infty }(\mathbb {T}^3)\) which is bounded in \([W^{1, \infty }(\mathbb {T}^3)]^{3}\).
By the results in [11], we can define
Let R be large enough, so that \(B(H^{2}, L^{\infty }, L^{\infty })\) is an absorbing ball of the system.
We next use these results directly, and introduce the classical smooth cut-off function \(\theta (r)\) in [37].
Express the original system as (1.2). It is worth highlighting that \(r(w)=\varphi (w)+h+g\), and here \(\varphi (v)\ge \beta _{1}>0\); especially, the abstract operator \(B(w)=\varphi (w)+h+2g\).
Define the cut-off function \(\theta (r)\in \mathcal {C^{\infty }}(\mathbb {R})\) as follows:
with \(|\theta '(r)|\le 2\).
Then, we replace \(\beta u\), \(\alpha v\), hw by \({\tilde{u}}\), \({\tilde{v}}\), \({\tilde{w}}\) defined as
so that
where R is defined in (5.2).
We can write (1.2) as
As in [31], let \(K_{2}>K_{1}>0\) and define the smooth functions \(\eta \) and \(\xi \) as follows:
obviously \(|\eta '|, |\xi '|<\infty \).
As in (4.1), the modified abstract differential system can be written as
on \(\Omega \) with periodic boundary conditions; the terms in the above system are defined as
Through these definitions, we know that F, G, \(\mathcal {H}\) are still globally Lipschitz continuous.
5.2 Verification of Conditions (1)–(5)
The hypotheses presented in Sect. 3 will be verified in several steps.
For all \(R_{1},R_{2}>0\), \(\mathcal {A},\mathcal {B},\mathcal {C},\mathcal {D}\) are defined as in (4.2).
(1) \(\mathbf {Regularity~ condition}:\)
For all \(R_{1},R_{2}>0\), the regularity condition holds. Through the above analysis, we can see that the weak Gateaux differentiability of F and G can be easily derived by the regularity of \({\tilde{u}}\). When \(u\in H^{2}\), we have \({\tilde{u}}\in H^{2}\), so that the regularity condition holds.
(2) \(\mathbf {Dissipativity~ condition}:\)
Let R, \(K_{2}\) be defined as in Sect. 5.1, and
Then the dissipativity condition holds.
When \(\Vert p\Vert>R_{1}>RK_{2}\), by definition of \(\eta \) and \(\xi \), we know that \( \xi (\Vert p\Vert ^{2})=0, \eta (\Vert Ap\Vert ^{2})=\frac{1}{2}\), so that \(\forall v\in L^{\infty }(\mathbb {T}^3)\) and we have
if \(v\in L^{\infty }(\mathbb {T}^3)\), \(\Vert v\Vert _{L^{\infty }(\mathbb {T}^3)}\ge R_{1}\), \(\xi (\Vert v\Vert ^{2}_{L^{\infty }(\mathbb {T}^3)})=0\), which means that \(G(p, 0, v)=0\). We can then derive that
In addition, if \(\Vert w\Vert _{L^{\infty }(\mathbb {T}^3)}\ge R_{1}\), \(\xi (\Vert w\Vert ^{2}_{L^{\infty }(\mathbb {T}^3)})=0\), for \(g<h\),
(3) \(\mathbf {Sobolev ~condition}:\)
We need to verify that for large \(\lambda _{N+1}\), \(\forall t\in [0, t_{0}]\), then \(\Vert Aq\Vert \le R_{2}\). By the above discussions and some important estimates in [31], we can get some important estimates on A,
Since G(p, 0, v) is bounded, we just need to show that \(\Vert q'(t)\Vert <\infty \). Next we start with the estimate on \(\Vert q\Vert \). For \(\Vert p \Vert \le R_{1}\), \(\xi (\Vert p \Vert ^{2})=1\).
Due to \(q(0)=0\), using the constant variation formula brings us to
so that
that is
By the Gronwall inequality,
that is \(\Vert q(t)\Vert \le \frac{4R}{\lambda _{N+1}}\).
Next, we compute \(\Vert q'(t)\Vert \); here \(\xi (\Vert p \Vert ^{2})=\xi (\Vert v\Vert ^{2}_{L^{\infty }(\mathbb {T}^3)})=1\).
Since
so that we have \(\Vert q'(0)\Vert \le 4R\), then
Taking the inner product with \(q'(t)\),
From the above assumptions, we have
Set \(|\xi '(\Vert v(t)\Vert ^{2}_{L^{\infty }(\mathbb {T}^3)})| < K_{1}\), so that
Let \(S=\frac{8RK_{1}R_{1}M}{\lambda }+4K_{1}R_{1}M(2R+R_{1})+2R+R_{1}\). By the Cauchy inequality,
which, together with the Gronwall inequality and \(\Vert q'(0)\Vert \le 4R\), yields
For \(\lambda \) large enough, let \(16R^{2}+\frac{S^{2}}{2(2\lambda -1)} \le R_{3}\). As a consequence, we obtain
which means that \(\Vert q'(t)\Vert ^{2}\) is bounded, so that the Sobolev condition holds.
Remark 5.1
We can also use the Bootstrap method to prove the Sobolev condition.
(4) \(\mathbf {Uniform ~cone ~condition}:\)
The verification of the uniform cone condition for the PDE part closely follows Theorem 3.1, achieved primarily through direct computations. The focus here is on confirming the subsequent spatial averaging principle:
Let \(\varphi \in H^1(\mathbb {T}^3)\). In the periodic situation, the Fourier series in Sect. 2 becomes the classical Fourier expansions. Let \(r>0\) and then set
First, a result concerning the eigenvalues of the negative Laplace operator under periodic boundary conditions is presented. Notably, in the given context, Lemma 2.1 remains applicable, allowing for a direct utilization of the subsequent lemma.
Lemma 5.1
Let \(\{ e_{m}\}^{\infty }_{m=1}\) denote a complete orthonormal set of eigenfunctions of \(-\triangle \) on \(\Omega \) with periodic boundary conditions. Then for any \( u\in H^{2}\), we have
where \(u_{m}=(u, e_{m})\) is the m-th Fourier coefficient.
Proof
By a simple computation, we have
This completes the proof. \(\square \)
Taking into account Lemma 2.1, we have
Using interpolation and with the help of Lemma 5.1, we have
Hence taking r large enough, \(\exists \delta >0\) such that
and the spatial averaging principle holds.
(5) \(\mathbf {Linear~ stability~ condition}:\)
It is imperative to confirm that \(F, G, \mathcal {H}\) maintain their Lipschitz continuity. Section 5.1 previously established that \(F, G, \mathcal {H}\) exhibit global Lipschitz continuity, ensuring the linear stability condition.
5.3 Existence of an Invariant Manifold
Theorem 5.3
Given the assumptions in Theorems 3.1 and 3.3, alongside conditions (1)–(5) from Sect. 4.2, when these conditions hold true, the coupled system possesses an invariant manifold, denoted by:
where \(\mathcal {P}=P_{N}(H^{1}\cap L^{\infty }(\mathbb {T}^3))\), \(\mathcal {Q}=Q_{N}(H^{1}\cap L^{\infty }(\mathbb {T}^3))\), \(\Psi \) is Lipschitz continuous, \(\mathcal {M}\) is an infinite-dimensional Lipschitz invariant manifold that enjoys the invariance property and the exponential tracking property.
Proof
Consider the original system, expressed as a coupled system comprising two differential equations. Represent the solution as \(y(t)=(u(t), v(t), w(t))\). This system is truncated into an absorbing ball, leading to a modified abstract differential system.
Define the semigroup of the original system as S(t),
and the semigroup of the modified abstract differential system as \({\tilde{S}}(t)\).
From the definition of the absorbing ball, \(\exists T>0\) such that for the initial value
we have \(\forall t\ge T,\)
which means that
Take \(R_{1}, R_{2}>R\). Then
here we still define
so that the invariance property holds. Owing to the results in Sects. 4.2 and 5.2, the desired result is obtained, concluding the proof. \(\square \)
Remark 5.2
Once we get an invariant manifold, we can reduce system (1.1) or (1.2) on this manifold and further analyze the dynamical behavior of the system, having in mind properties such as attractor bifurcation and hopf bifurcations. We can also investigate the local dynamical behavior of the traveling wave or spike-like solutions of the system, construct stable manifolds, unstable manifolds, etc. These questions are all worthy of further study, as noted in [32], some of which still being open problems.
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Yan, X., Yin, K., Yang, XG. et al. Invariant Manifolds for a PDE-ODE Coupled System. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10353-y
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DOI: https://doi.org/10.1007/s10884-024-10353-y