Open Access

\(H^{2}\)-boundedness of the pullback attractor of the micropolar fluid flows with infinite delays

Boundary Value Problems20172017:133

https://doi.org/10.1186/s13661-017-0866-x

Received: 6 April 2017

Accepted: 1 September 2017

Published: 13 September 2017

Abstract

We establish the \(H^{2}\)-boundedness of the pullback attractor for a two-dimensional nonautonomous micropolar fluid flow with infinite delays.

Keywords

micropolar fluid flowPullback attractor \(H^{2}\)-boundednessinfinite delays

MSC

5B4135B6535Q35

1 Introduction

The 3D micropolar fluid model was firstly formulated by Eringen [1] and was used to describe the fluids consisting of randomly oriented particles suspended in a viscous medium. According to [1], the incompressible micropolar fluid motion can be expressed by the following system:
$$\begin{aligned} \textstyle\begin{cases} \frac{\partial u}{\partial t} -(\nu+\nu_{r})\Delta u -2\nu_{r}\nabla\times\omega +(u \cdot\nabla)u +\nabla p =f, \\ \nabla\cdot u =0, \\ \frac{\partial\omega}{\partial t} -(c_{a}+c_{d})\Delta\omega +4\nu_{r} \omega +(u \cdot\nabla)\omega -(c_{0}+c_{d}-c_{a})\nabla(\nabla\cdot\omega) -2\nu_{r} \nabla\times u =\tilde{f}, \end{cases}\displaystyle \end{aligned}$$
where \(u(x,t)=(u_{1},u_{2},u_{3})\) represents the velocity, \(\omega (x,t)=(\omega_{1},\omega_{2},\omega_{3})\) stands for the angular velocity field of rotation of particles, p is the pressure, f and represent the external force and moment, respectively. The positive parameters ν, \(\nu_{r}\), \(c_{0}\), \(c_{a}\), \(c_{d}\) are the viscous coefficients. In fact, ν is the usual Newtonian kinetic viscosity, and \(\nu_{r}\) is the dynamics microrotation viscosity, and \(c_{0}\), \(c_{a}\), \(c_{d}\) denote the angular viscosity (see [2]). From [1, 2] we see that these equations express the balance of momentum, mass, and moment of momentum, accordingly. When microrotation effects are neglected (i.e., \(\omega =0\)), the equations reduce to the incompressible Navier-Stokes equations. Therefore, the equations of micropolar fluid flows can be regarded as a generalization of the Navier-Stokes equations in the sense that they take into account the microstructure of the fluid. For physical background, we refer, for example, to [2, 3].

Due to their wide applications, the micropolar fluid flows have drawn much attention from mathematicians and physicists and have been well studied. For the theories on the existence and uniqueness of solutions of the micropolar fluid flows, we refer to [39]. At the same time, the long-time behavior of solutions for the micropolar fluid flows has been investigated from various aspects. Chen et al. proved the existence of \(H^{2}\)-compact global attractors in a bounded domain [10] and verified the existence of uniform attractors in nonsmooth domains [11]. Lukaszewicz [12] established the existence of \(H^{1}\)-pullback attractor for nonautonomous micropolar fluid flows in a bounded domain. As for the long-time behavior of solutions for the micropolar fluid flows on unbounded domains, Dong and Chen [13] discussed the existence and regularity of the global attractors. Later, they [14] obtained the \(L^{2}\) time decay rate for global solutions of the 2D micropolar equations via the Fourier splitting method. Chen and Price [15] obtained the \(L^{2}\) time decay rate for small solutions of the 3D micropolar equations via Kato’s method. Zhao et al. [16] showed the existence of an \(H^{1}\)-uniform attractor and so on. For more theories about the micropolar fluid flows, we refer to [1721].

There are also some efforts focused on the 2D micropolar equations with partial dissipation. Dong and Zhang [22] examined the microrotation viscosity, namely \(c_{a}+c_{d}=0\). The global regularity problem for this partial dissipation case is not trivial due to the presence of the term \(\nabla\times\omega\) in the velocity equation. Dong and Zhang overcame the difficulty by making full use of the quantity \(\nabla\times u -\frac{2\nu_{r}}{\nu+\nu_{r}}\omega\), which obeys a transport-diffusion equation. When the parameters \(\nu =0\) and \(\nu_{r}\neq c_{a}+c_{d}\), the global well-posedness of the micropolar fluid equations were obtained in the framework of Besov spaces [23]. More recently, Dong et al. [24] studied the global regularity and large-time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation, in which they established the well-posedness of the solutions by fully exploiting the structure of the system and controlling the vorticity via the evolution equation of a combined quantity of the vorticity and the microrotation angular velocity; they also obtained suitable decay rates of the solution by combining diagonalization process with uniformly bounded estimates for the first derivatives of the solutions.

In this paper, we consider the special situation where the velocity component in the \(x_{3}\)-direction is zero and the axes of rotation of particles are parallel to the \(x_{3}\)-axis, that is, \(u=(u_{1},u_{2},0), \omega=(0,0,\omega_{3}), f=(f_{1},f_{2},0)\), and \(\tilde{f}=(0,0,\tilde{f}_{3})\). Let \(\Omega\subset\mathbb{R}^{2}\) be a bounded open domain with smooth boundary Ω such that the following Poincaré inequality holds:
$$\begin{aligned} \mbox{There exists } \lambda_{1}>0 \quad\mbox{such that} \quad\lambda_{1}\Vert \varphi \Vert ^{2}_{L^{2}(\Omega)} \leqslant \Vert \nabla\varphi \Vert ^{2}_{L^{2}(\Omega)},\quad \forall\varphi\in H_{0}^{1}(\Omega). \end{aligned}$$
(1.1)
Then, we discuss the following 2D non-autonomous incompressible micropolar fluid flows with infinite delays in Ω:
$$ \textstyle\begin{cases} \frac{\partial u}{\partial t} -(\nu+\nu_{r})\Delta u -2\nu_{r}\mathrm{\nabla}\times\omega +(u \cdot\nabla)u+\nabla p\\ \quad=f(t,x)+g(t, u_{t}),& t>\tau, x\in\Omega, \\ \frac{\partial\omega}{\partial t} -\alpha\Delta\omega +4\nu_{r} \omega -2\nu_{r} \nabla\times u +(u \cdot\nabla)\omega \\ \quad=\tilde{f}(t,x)+\tilde{g}(t, \omega_{t}),& t>\tau, x\in\Omega, \\ \nabla\cdot u=0,& \mbox{in } (\tau, +\infty)\times\Omega, \\ u=0, \qquad\omega=0,& \mbox{on } (\tau, +\infty)\times{\partial \Omega}, \\ (u(\tau+s, x), \omega(\tau+s, x))=\phi(s, x),& s\in(-\infty, 0],\tau\in{\mathbb{R}}, x\in\Omega, \end{cases} $$
(1.2)
where \(x=(x_{1},x_{2})\in\Omega\) and \(\alpha=c_{a}+c_{d}\). The vector functions \(g=(g_{1},g_{2},0)\) and \(\tilde{g}=(0,0,\tilde {g}_{3})\) are additional external forces containing some hereditary characteristics \(u_{t}\) and \(\omega_{t}\), which are defined on \((-\infty, 0]\) as follows:
$$ u_{t}=u_{t}(\cdot):=u(t+\cdot), \qquad\omega _{t}=\omega_{t}(\cdot):=\omega(t+\cdot),\quad t\geqslant \tau. $$
(1.3)
In addition, \(\phi(s,x)=(u_{\tau},\omega_{\tau})=(u(\tau+s, x), \omega(\tau+s, x))\) is the initial datum in the interval of delay time \((-\infty,0]\), and
$$ \nabla\times u:=\frac{\partial u_{2}}{\partial x_{1}}-\frac{\partial u_{1}}{\partial x_{2}} \quad\mbox{and}\quad \nabla\times\omega:=\biggl(\frac{\partial\omega}{\partial x_{2}}, -\frac{\partial\omega}{\partial x_{1}}\biggr). $$

In the real world, delay terms appear naturally, for instance, as effects in wind tunnel experiments. Also, the delay situations may occur when we want to control the system via applying a force that considers not only the present state but also the history state of the system. However, so far, to our knowledge, there is no references discussing the micropolar fluid flows with delay in addition to [25], where the author established the global well-posedness and pullback attractors for a 2D impressible micropolar flows with infinite delays.

The main purpose of this work is to establish the \(H^{2}\)-boundedness of the pullback attractor \(\widehat{\mathcal{A}}= \{\mathcal{A}(t) \mid t\in\mathbb{R} \}\) obtained in [25]. Before stating the main results of this paper, we give some assumptions:
(A1)
(I)
A mapping \(G: [\tau, T]\times\mathcal {C}_{\gamma}(\widehat{H}) \mapsto(L^{2}(\Omega))^{3}\) satisfies:
  1. (i)

    For any \(\xi\in\mathcal{C}_{\gamma}(\widehat{H})\), the mapping \([\tau, T]\ni t \mapsto G(t,\xi)\in(L^{2}(\Omega))^{3}\) is measurable;

     
  2. (ii)

    \(G(\cdot, 0)=(0,0,0)\);

     
  3. (iii)
    There exists a constant \(L_{G}>0\) such that, for any \(t\in[\tau, T]\) and \(\xi, \eta\in\mathcal{C}_{\gamma}(\widehat{H})\),
    $$ \bigl\Vert G(t,\xi)-G(t,\eta) \bigr\Vert \leqslant L_{G}\Vert \xi-\eta \Vert _{\gamma}. $$
     
(II)
\(F(t,x)\in L_{\mathrm{loc}}^{2}(\mathbb{R}; \widehat{H}), 2L_{G}<\delta_{1}\lambda_{1}<2\gamma\), and
$$ \int_{\tau}^{t} e^{(\delta_{1}\lambda_{1}-2L_{G})(r-\tau)}\bigl\Vert F(r) \bigr\Vert ^{2}\,{\mathrm{d}}r < + \infty,\quad\forall\tau\in\mathbb{R}, t\geqslant\tau. $$
(A2)
(I)
\(\frac{\mathrm{d}G}{\mathrm{d}t}:=(G(t,\xi))' : [\tau, T]\times \mathcal{C}_{\gamma}(\widehat{H})\mapsto(L^{2}(\Omega))^{3}\) satisfies:
  1. (i)

    For any \(\xi(t), \xi'(t)\in\mathcal{C}_{\gamma}(\widehat {H})\), the mapping \(t \mapsto(G(t,\xi))'\) is measurable;

     
  2. (ii)

    \((G(\cdot, 0))'=(0,0,0)\);

     
  3. (iii)
    There exists a constant \(\tilde{L}_{G}>0\) such that, for any \(t\in[\tau, T]\), \(\xi'(t), \eta'(t) \in\mathcal{C}_{\gamma }(\widehat{H})\),
    $$ \bigl\Vert \bigl(G(t,\xi)\bigr)'-\bigl(G(t,\eta) \bigr)' \bigr\Vert \leq\tilde{L}_{G}\bigl\Vert \xi '-\eta' \bigr\Vert _{\gamma}. $$
     
(II)
\(F(t,x)\in W_{\mathrm{loc}}^{1,2}(\mathbb{R}; \widehat{H}), 2\tilde{L}_{G}<\delta_{1}\lambda_{1}<2\gamma\), and
$$ \int_{\tau}^{t} e^{(\delta_{1}\lambda_{1}-2\tilde{L}_{G})(r-\tau )}\bigl\Vert F'(r) \bigr\Vert ^{2}\,{\mathrm{d}}r < + \infty,\quad \forall \tau\in\mathbb{R}, t\geqslant\tau. $$

Under the above assumptions, we have

Theorem 1.1

Assume that (A1) and (A2) hold.
  1. (1)

    For any bounded set \(\mathcal{B}\subset\mathcal{C}_{\gamma }(\widehat{H})\) and any \(\tau\in\mathbb{R}, \epsilon>0, t\geqslant\tau+2\epsilon \), the set \(\bigcup_{s\in[\tau+2\epsilon,t]} U(s,\tau)\mathcal{B}\) is bounded in \(D(A)=\widehat{V}\cap(H^{2}(\Omega))^{3}\).

     
  2. (2)

    Let \(\widehat{\mathcal{A}}= \{\mathcal{A}(t) \mid t\in\mathbb {R} \}\) be the pullback attractor of system (1.2). Then, for any \(T_{1}, T_{2} \in\mathbb{R}\) with \(T_{1}< T_{2}\), the set \(\bigcup _{t\in[T_{1},T_{2}]}\mathcal{A}(t)\) is bounded in \(D(A)=\widehat{V}\cap(H^{2}(\Omega))^{3}\).

     

We remark that García-Luengo et al. [9] proved the existence of the pullback attractor and investigated its tempered behavior for Navier-Stokes equations in bounded domains. Further, they discussed the \(H^{2}\)-boundedness of the pullback attractors of the Navier-Stokes equations in [26]. Recently, Zhao and Sun [25] established the existence of pullback attractors for 2D nonautonomous micropolar fluid flows with infinite delays. Motivated by [26] and following its main idea, we generalize their results to the micropolar fluid flows with infinite delays. Compared with the Navier-Stokes equations (\(\omega=0, \nu_{r}=0\)), the micropolar fluid flow consists of the angular velocity field ω, which leads to a different nonlinear term \(B(u,w)\) and an additional term \(N(u)\) in the abstract equation. In addition, the time-delay term considered in this work also increases the difficulty. Therefore, we have to obtain more delicate estimates and analysis for the solutions.

The paper is organized as follows. In Section 2, we make some preliminaries. That is, we introduce some notations and recall some known results. In Section 3, we concentrate on showing the \(H^{2}\)-boundedness of the pullback attractor \(\widehat{\mathcal{A}}\). To this end, we first make some estimates for the Galerkin approximation solutions by mainly using the energy method. Then, we obtain a general result about \(\widehat{V}\cap(H^{2}(\Omega ))^{3}\)-boundedness of invariant sets for the associate evolution process. Further, we have the boundedness of the pullback attractor in \(\widehat{V}\cap(H^{2}(\Omega))^{3}\).

2 Preliminaries

In this section, we make some necessary preliminaries by introducing some notation and key operators. Then, we rewrite equations (1.2) in an abstract form. Finally, we recall some known results.

We denote by \(L^{p}(\Omega)\) and \(W^{m,p}(\Omega)\) the usual Lebesgue and Sobolev spaces (see [27]) endowed with norms \(\Vert \cdot \Vert _{p}\) and \(\Vert \cdot \Vert _{m,p}\), respectively:
$$\begin{aligned} \Vert \varphi \Vert _{p}:= \biggl( \int_{\Omega} \vert \varphi \vert ^{p}\,{\mathrm{d}}x \biggr)^{1/p}\quad\mbox{and}\quad \Vert \varphi \Vert _{m,p}:=\biggl(\sum_{\vert \beta \vert \leqslant m} \int_{\Omega}\bigl\vert D^{\beta}\varphi\bigr\vert ^{p}\,{\mathrm{d}}x\biggr)^{1/p}. \end{aligned}$$
In particular, we denote \(H^{m}(\Omega):=W^{m,2}(\Omega)\) and by \(H_{0}^{1}(\Omega)\) the closure of \(\mathcal{C}_{0}^{\infty}(\Omega )\) with respect to the \(H^{1}(\Omega)\) norm.
$$\begin{aligned} &\mathcal{V} := \bigl\{ \varphi\in\mathcal{C}_{0}^{\infty}( \Omega) \times\mathcal{C}_{0}^{\infty}(\Omega)\mid\varphi=( \varphi_{1}, \varphi_{2}), \nabla\cdot\varphi=0 \bigr\} , \\ & H := \mbox{closure of } \mathcal{V} \mbox{ in } L^{2}(\Omega)\times L^{2}(\Omega) \mbox{ with norm }\Vert \cdot \Vert _{H} \mbox{ and dual space} H^{*}, \\ &V := \mbox{closure of } \mathcal{V} \mbox{ in } H^{1}(\Omega)\times H^{1}(\Omega) \mbox{ with norm } \Vert \cdot \Vert _{V} \mbox{ and dual space } V^{*}, \\ & \widehat{H} := H \times L^{2}(\Omega) \mbox{ with norm } \Vert \cdot \Vert _{\widehat{H}} \mbox{ and dual space } \widehat{H}^{*}, \\ & \widehat{V} := V \times H_{0}^{1}(\Omega) \mbox{ with norm } \Vert \cdot \Vert _{\widehat{V}} \mbox{ and dual space } \widehat{V}^{*}, \end{aligned}$$
where \(\Vert \cdot \Vert _{H}\), \(\Vert \cdot \Vert _{V}\), \(\Vert \cdot \Vert _{\widehat{H}}\), and \(\Vert \cdot \Vert _{\widehat{V}}\) are defined by
$$\begin{aligned} &\bigl\Vert (u,v) \bigr\Vert _{H}:=\bigl(\Vert u \Vert _{2}^{2}+\Vert v \Vert _{2}^{2} \bigr)^{1/2}, \\ &\bigl\Vert (u,v) \bigr\Vert _{V}:=\bigl(\Vert u \Vert ^{2}_{H^{1}}+\Vert v \Vert ^{2}_{H^{1}}\bigr)^{1/2}, \\ &\bigl\Vert (u,v,w) \bigr\Vert _{\widehat{H}}:=\bigl(\bigl\Vert (u,v) \bigr\Vert ^{2}_{H}+\Vert w \Vert _{2}^{2} \bigr)^{1/2}, \\ & \bigl\Vert (u,v,w) \bigr\Vert _{\widehat{V}}:= \bigl(\bigl\Vert (u,v) \bigr\Vert ^{2}_{V}+\Vert w \Vert ^{2}_{H^{1}}\bigr)^{1/2}; \end{aligned}$$
\((\cdot,\cdot) \) is the inner product in \(L^{2}(\Omega)\), H, or Ĥ, and \(\langle\cdot, \cdot\rangle\) is the dual pairing between V and \(V^{*}\) or between and \(\widehat{V}^{*}\). Throughout this article, we simplify the notations \(\Vert \cdot \Vert _{2}\), \(\Vert \cdot \Vert _{H}\), and \(\Vert \cdot \Vert _{\widehat {H}}\) by the same notation \(\Vert \cdot \Vert \) if there is no confusion. Furthermore, we denote
$$\begin{aligned}& \mathcal{C}_{\gamma}(\widehat{H}) := \Bigl\{ \varphi\in\mathcal{C}\bigl((-\infty,0]; \widehat{H}\bigr)\mid\exists\lim_{s\rightarrow-\infty} e^{\gamma s} \varphi(s)\in\widehat{H} \Bigr\} \mbox{ with some suitable } \gamma>0, \\& \hphantom{\mathcal{C}_{\gamma}(\widehat{H}) := } \mbox{which is a Banach space with the norm} \\& \hphantom{\mathcal{C}_{\gamma}(\widehat{H}) := } \Vert \varphi \Vert _{\gamma} := \sup_{s\in(-\infty,0]} e^{\gamma s}\bigl\Vert \varphi(s) \bigr\Vert ; \\& L^{p}(I; X) := \mbox{space of strongly measurable functions on the closed interval $I$} \\& \hphantom{L^{p}(I; X) := }\mbox{ with values in a Banach space $X$, endowed with norm} \\& \hphantom{L^{p}(I; X) := } \Vert \varphi \Vert _{L^{p}(I; X)}:=\biggl(\int_{I} \Vert \varphi \Vert ^{p}_{X}\,{\mathrm{d}}t\biggr)^{1/p} \quad\mbox{for }1\leqslant p< \infty, \\& \mathcal{C}(I; X) :=\mbox{space of continuous functions on the interval $I$ with values} \\& \hphantom{\mathcal{C}(I; X) :=}\mbox{ in the Banach space $X$, endowed with the usual norm,} \\& L_{\mathrm{loc}}^{2}(I;\widehat{H}) :=\mbox{space of locally integrable functions from the interval $I$ to $\widehat{H}$}, \\& \operatorname{dist}_{M}(X,Y) \mbox{ is the Hausdorff semidistance between } X \subseteq M \mbox{ and } Y \subseteq M \mbox{ defined by} \\& \operatorname{dist}_{M}(X,Y)=\sup_{x\in X}\inf_{y\in Y} \operatorname{dist}_{M}(x,y). \end{aligned}$$
Now, we introduce three operators:
$$\begin{aligned} &\langle Aw, \varphi\rangle:= (\nu+\nu_{r}) (\nabla u, \nabla \Psi) +\alpha(\nabla\omega, \nabla\psi), \quad \forall w =(u,\omega)\in \widehat{V}, \forall\varphi=(\Psi, \psi)\in\widehat{V}, \\ &\bigl\langle B(u, w), \varphi\bigr\rangle := \bigl((u \cdot\nabla )w, \varphi \bigr), \quad \forall u\in V, w\in\widehat{V}, \forall\varphi\in\widehat{V}, \\ &N(w) :=(-2\nu_{r} \nabla\times\omega, -2\nu_{r} \nabla\times u+4\nu_{r}\omega), \quad \forall w=(u, \omega)\in \widehat{V}. \end{aligned}$$
There are some useful estimations for the operators A, \(B(\cdot ,\cdot)\), and \(N(\cdot)\) established in [3, 25, 28].

Lemma 2.1

  1. (1)
    The operator A is linear continuous both from to \(\widehat{V}^{*}\) and from \(D(A)=\widehat{V}\cap(H^{2}(\Omega))^{3}\) to Ĥ. Moreover, there are two positive constants \(c_{1}\) and \(c_{2}\) such that
    $$\begin{aligned} c_{1}\langle Aw, w \rangle\leqslant \Vert w \Vert _{\widehat{V}}^{2} \leqslant c_{2}\langle Aw, w \rangle, \quad\forall w\in{\widehat{V}}. \end{aligned}$$
    (2.1)
    In addition, for any \(w\in D(A)\), we have
    $$\begin{aligned} \delta \Vert \nabla w \Vert ^{2} \leqslant\langle Aw, w \rangle\leqslant \Vert w \Vert \Vert Aw \Vert \leqslant \lambda _{1}^{-\frac{1}{2}} \Vert \nabla w \Vert \Vert Aw \Vert , \end{aligned}$$
    (2.2)
    where \(\delta= \min\{\nu+\nu_{r}, \alpha\}\), and \(\lambda_{1}\) is the constant from (1.1).
     
  2. (2)
    The operator \(B(\cdot,\cdot)\) is continuous from \(V\times\widehat {V}\) to \(\widehat{V}^{*}\) and satisfies the following properties:
    1. (i)
      For any \(u\in V\) and \(w\in\widehat{V}\), we have
      $$\begin{aligned} \bigl\langle B(u,w), \varphi\bigr\rangle = - \bigl\langle B(u, \varphi), w \bigr\rangle , \quad\forall\varphi\in\widehat{V}. \end{aligned}$$
      (2.3)
      In particular,
      $$\begin{aligned} \bigl\langle B(u,w), w \bigr\rangle = 0, \quad\forall u\in V, w\in \widehat{V}. \end{aligned}$$
      (2.4)
       
    2. (ii)
      There exists a positive constant λ, which depends only on Ω, such that for any \((u, \psi, \varphi)\in V\times\widehat{V}\times\widehat{V}\), we have
      $$\begin{aligned} \bigl\vert \bigl\langle B(u,\psi),\varphi\bigr\rangle \bigr\vert \leqslant \textstyle\begin{cases} \lambda \Vert u \Vert ^{\frac{1}{2}}\Vert \nabla u \Vert ^{\frac{1}{2}} \Vert \varphi \Vert ^{\frac{1}{2}}\Vert \nabla\varphi \Vert ^{\frac{1}{2}}\Vert \nabla\psi \Vert , \\ \lambda \Vert u \Vert ^{\frac{1}{2}}\Vert \nabla u \Vert ^{\frac{1}{2}} \Vert \psi \Vert ^{\frac{1}{2}}\Vert \nabla\psi \Vert ^{\frac {1}{2}}\Vert \nabla\varphi \Vert . \end{cases}\displaystyle \end{aligned}$$
      (2.5)
      Moreover, if \((u, \psi, \varphi)\in V\times D(A)\times D(A)\), then
      $$ \bigl\vert \bigl\langle B(u,\psi),A\varphi\bigr\rangle \bigr\vert \leqslant\lambda \Vert u \Vert ^{\frac{1}{2}}\Vert \nabla u \Vert ^{\frac{1}{2}} \Vert \nabla\psi \Vert ^{\frac{1}{2}}\Vert A\psi \Vert ^{\frac{1}{2}} \Vert A\varphi \Vert . $$
      (2.6)
       
     
  3. (3)
    The operator \(N(\cdot)\) is continuous from to Ĥ. Moreover, there exists a positive constant \(c(\nu_{r})\) such that
    $$\begin{aligned} \bigl\Vert N(\psi) \bigr\Vert \leqslant c(\nu _{r})\Vert \psi \Vert _{\widehat{V}}, \quad\forall\psi\in \widehat{V}. \end{aligned}$$
    (2.7)
    In addition,
    $$\begin{aligned} &{-}\bigl\langle N(\psi), A\psi\bigr\rangle \leqslant\frac{1}{4} \Vert A\psi \Vert ^{2}+c^{2}(\nu_{r})\Vert \psi \Vert _{\widehat{V}}^{2},\quad\forall\psi\in D(A), \end{aligned}$$
    (2.8)
    $$\begin{aligned} &\delta_{1}\Vert \psi \Vert _{\widehat{V}}^{2} \leqslant\langle A\psi, \psi\rangle+\bigl\langle N(\psi), \psi \bigr\rangle , \quad\forall\psi\in\widehat{V}; \end{aligned}$$
    (2.9)
    hereinafter \(\delta_{1} :=\min\{\nu, \alpha\}\).
     
According to the previous notation, we can formulate a weak version of system (1.2) as follows:
$$\begin{aligned} \textstyle\begin{cases} \frac{\partial w}{\partial t} + Aw + B(u,w) + N(w) = F(t,x)+G(t,w_{t}),& t>\tau, \\ w|_{t=\tau}=w_{\tau}=(u_{\tau},w_{\tau})=(u(\tau+s),\omega(\tau+s)) :=\phi(s),& s\in(-\infty, 0], \end{cases}\displaystyle \end{aligned}$$
(2.10)
where \(w=(u,\omega), F(t)=F(t,x):=(f(t,x),\tilde{f}(t,x))\), and \(G(t,w_{t}):=(g(t,u_{t}),\tilde{g}(t,\omega_{t}))\).
We say that a function \(w\in\mathcal{C}((-\infty,T];\widehat{H})\cap L^{2}(\tau,T;\widehat{V})\) with \(w_{\tau}=\phi(s) \in\mathcal{C}_{\gamma}(\widehat{H})\) is a weak solution of system (2.10) in the interval \((-\infty,T]\) if, for all \(T>\tau\) and \(\varphi\in\widehat{V}\), the following equation holds in the distribution sense of \(\mathcal{D}'(\tau,T)\):
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(w,\varphi) + \langle Aw, \varphi \rangle+ \bigl\langle B(u,w), \varphi\bigr\rangle + \bigl\langle N(w), \varphi\bigr\rangle = \bigl\langle F(t), \varphi\bigr\rangle + \bigl(G(t,w_{t}),\varphi\bigr). \end{aligned}$$

Lemma 2.2

see [25]

Assume that (A1) holds. Then for any given initial datum \(w_{\tau}:=\phi(s)\in\mathcal {C}_{\gamma}(\widehat{H})\) and any \(T>\tau\), there exists a unique stable weak solution
$$ w\in\mathcal{C}\bigl((-\infty,T);\widehat{H}\bigr)\cap L^{2}(\tau, T; \widehat{V}), \quad w'\in L^{2}\bigl(\tau,T; \widehat{V}^{*}\bigr). $$
Moreover, for any \(t\in[\tau,T]\),
$$\begin{aligned}& \Vert w_{t} \Vert _{\gamma}^{2} \leqslant e^{(-\delta_{1}\lambda _{1}+2L_{G})(t-\tau)}\bigl\Vert \phi(s) \bigr\Vert _{\gamma}^{2} + \frac{2}{\delta_{1}} \int_{\tau}^{t} e^{(-\delta_{1}\lambda_{1}+2L_{G})(t-\theta)} \bigl\Vert F(\theta ) \bigr\Vert ^{2}\,\mathrm{d}\theta, \end{aligned}$$
(2.11)
$$\begin{aligned}& \delta_{1}\int_{\tau}^{t} \bigl\Vert w(\theta) \bigr\Vert _{\widehat{V}}^{2}\,\mathrm{d}\theta\leqslant 2e^{\delta_{1}\lambda _{1}(t-\tau)}\bigl\Vert w(\tau) \bigr\Vert ^{2} + \frac{4}{\delta_{1}}e^{2L_{G} t-\delta_{1}\lambda_{1}\tau} \int_{\tau}^{t} e^{(\delta_{1}\lambda_{1}-2L_{G})\theta} \bigl\Vert F(\theta ) \bigr\Vert ^{2}\,\mathrm{d}\theta \\& \hphantom{\delta_{1}\int_{\tau}^{t} \bigl\Vert w(\theta) \bigr\Vert _{\widehat{V}}^{2}\,\mathrm{d}\theta\leqslant}+\frac{5}{\delta _{1}}e^{-\delta_{1}\lambda_{1}\tau} \int_{\tau}^{t} e^{\delta_{1}\lambda_{1} \theta} \bigl\Vert F(\theta ) \bigr\Vert ^{2}\,\mathrm{d}\theta+ 2e^{2L_{G}(t-\tau)}\bigl\Vert \phi(s) \bigr\Vert _{\gamma}^{2}. \end{aligned}$$
(2.12)
In addition, if \(w_{\tau}\in\widehat{V}\), then the weak solution \(w\in\mathcal{C}((-\infty,T);\widehat{V})\cap L^{2}(\tau,T;D(A))\).
Based on Lemma 2.2, we can define the map
$$\begin{aligned} U(t,\tau): w_{\tau}(\cdot):=\phi(s)\mapsto U(t,\tau ;w_{\tau}) =U(t,\tau)\phi(s)= w_{t}(\cdot),\quad t\geqslant\tau , s\in(-\infty,0], \end{aligned}$$
(2.13)
which generates a continuous process in \(\mathcal{C}_{\gamma}(\widehat {H})\) satisfying:
  • \(U(s,s) = \mbox{identity}\),

  • \(U(t,r)U(r,s) = U(t,s) \mbox{ for any } s \leqslant r \leqslant t\),

where w is the solution of system (2.10) corresponding to the initial datum \(\phi(s)\in{\mathcal{C}}_{\gamma}(\widehat{H})\), and \(w_{t}(s)\) is defined as in (1.3).

Lemma 2.3

Under assumption (A1), there exists a pullback attractor \(\widehat{\mathcal{A}}= \{ \mathcal{A}(t)\mid t\in\mathbb{R} \}\) for the process \(\{U(t,\tau)\}_{t\geqslant\tau}\) that satisfies the following properties:
  • Compactness: for any \(t\in\mathbb{R}\), \(\mathcal{A}(t)\) is a nonempty compact subset of \(\mathcal{C}_{\gamma}(\widehat{H})\);

  • Invariance: \(U(t, \tau)\mathcal{A}(\tau)=\mathcal{A}(t)\), \(\forall t\geqslant\tau\);

  • Pullback attracting: for any bounded set \(\mathcal{B}\) of \(\mathcal{C}_{\gamma}(\widehat{H})\), we have
    $$\begin{aligned} \lim_{\tau\rightarrow-\infty} \operatorname{dist}_{ \mathcal {C}_{\gamma}(\widehat{H})} \bigl( U(t, \tau) \mathcal{B}, \mathcal{A}(t) \bigr) =0, \quad\forall t\in\mathbb{R}. \end{aligned}$$

Now, we end this section with the following lemma, which plays an important role in the proof of higher regularity of the pullback attractors.

Lemma 2.4

 (see [29, 30]) Let X, Y be Banach spaces such that X is reflexive and the inclusion \(X\subset Y\) is continuous. Assume that \(\{w_{n}\}_{n\geqslant1}\) is a bounded sequence in \(L^{\infty}(\tau,t; X)\) such that \(w_{n}\rightharpoonup w\) weakly in \(L^{q}(\tau,t; X)\) for some \(q\in[1,+\infty)\) and \(w\in\mathcal {C}([\tau,t]; Y)\). Then \(w(t)\in X\), and
$$\begin{aligned} \bigl\Vert w(s) \bigr\Vert _{X} \leqslant\liminf _{n\rightarrow+\infty}\bigl\Vert w_{n}(s) \bigr\Vert _{L^{\infty}(\tau,t; X)}, \quad \forall s\in[\tau,t]. \end{aligned}$$

3 \(H^{2}\)-boundedness of the pullback attractor

In this section, we concentrate on proving the \(H^{2}\)-boundedness of the pullback attractor \(\widehat{\mathcal{A}}\).

To begin with, let us recall some properties of the operator A. According to the classical spectral theory of elliptic operators (see [31]), there exists a sequence \(\{\lambda_{n}\}_{n=1}^{\infty}\) satisfying
$$ 0< \lambda_{1} \leqslant\lambda_{2} \leqslant\cdots \leqslant\lambda_{n} \leqslant\cdots, \quad\lambda_{n} \rightarrow+\infty\mbox{ as } n \rightarrow\infty, $$
and a sequence of elements \(\{v_{n}\}_{n=1}^{\infty}\subseteq D(A)\) forming a Hilbert basis of Ĥ and such that the span of \(\{v_{1},v_{2},\ldots,v_{n},\ldots\}\) is dense in and
$$\begin{aligned} Av_{n}=\lambda_{n} v_{n},\quad \forall n\in\mathbf{N}. \end{aligned}$$
(3.1)
For each \(T>\tau\), denote by \(w^{(m)}(t)=w^{(m)}(t;\tau,w_{\tau}):=\sum_{j=1}^{m} \beta _{m,j}(t)v_{j}\) the Galerkin approximation solutions of the solution \(w(t)\) of system (2.10), which is the solution of the following ordinary differential equations:
$$\begin{aligned} & \frac{\mathrm{d}}{\mathrm{d}t}\bigl(w^{(m)}(t),v_{j}\bigr) +\bigl\langle Aw^{(m)}(t),v_{j} \bigr\rangle +\bigl\langle B \bigl(u^{(m)}(t),w^{(m)}(t)\bigr),v_{j} \bigr\rangle +\bigl\langle N\bigl(w^{(m)}(t)\bigr),v_{j} \bigr\rangle \\ & \quad = \bigl\langle F(t),v_{j} \bigr\rangle + \bigl(G\bigl(t,w_{t}^{(m)}\bigr),v_{j}\bigr), \quad 1\leqslant j\leqslant m, t\in(\tau,T), \end{aligned}$$
(3.2)
$$\begin{aligned} & w^{(m)}_{\tau}(s) = w^{(m)}(\tau+s) =P_{m}\phi(s), \quad s\in(-\infty,0]. \end{aligned}$$
(3.3)
Now, we verify the following results about the Galerkin approximation solutions.

Lemma 3.1

Assume that (A1) holds. Then, for any bounded subset \(\mathcal {B}\) of \(\mathcal{C}_{\gamma}(\widehat{H})\) and any \(\epsilon>0, \tau \in\mathbb{R}, t>\tau+\epsilon\), we have that
  1. (i)

    the set \(\{ w^{(m)}(\theta; \tau,w_{\tau}) \mid\theta \in[\tau+\epsilon, t], w_{\tau}:=\phi(s)\in\mathcal{B} \}\) is bounded in ,

     
  2. (ii)

    the set \(\{ w^{(m)}(\cdot; \tau,w_{\tau}) \mid w_{\tau }\in\mathcal{B} \}\) is bounded in \(L^{2}(\tau+\epsilon,t; D(A))\), and

     
  3. (iii)

    the set \(\{ w^{(m)'}(\cdot; \tau,w_{\tau}) \mid w_{\tau}\in\mathcal{B} \}\) is bounded in \(L^{2}(\tau+\epsilon,t; \widehat{H})\), where \(w^{(m)'}(\theta)=\frac{\mathrm{d}w^{m}(\theta)}{\mathrm{d}\theta}\).

     

Proof

For any fixed bounded set \(\mathcal{B}\subset\mathcal{C}_{\gamma }(\widehat{H}), \tau\in\mathbb{R}, \epsilon>0, t>\tau+\epsilon\), and \(w_{\tau}=\phi(s)\in\mathcal{C}_{\gamma}(\widehat{H})\) multiplying (3.2) by \(\beta_{m,j}(t)\), summing up for j from 1 to m, and then using (2.4) and (2.9), we obtain
$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\theta} \bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2} + \delta_{1}\bigl\Vert w^{(m)}( \theta) \bigr\Vert _{\widehat{V}}^{2} \\ &\quad\leqslant \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\theta}\bigl\Vert w^{(m)}( \theta) \bigr\Vert ^{2} + \bigl\langle Aw^{(m)}(\theta), w^{(m)}(\theta) \bigr\rangle + \bigl\langle N\bigl(w^{(m)} \bigr), w^{(m)}(\theta) \bigr\rangle \\ &\quad= \bigl(F(\theta), w^{(m)}(\theta)\bigr) + \bigl(G \bigl(t,w_{\theta}^{(m)}\bigr),w^{(m)}\bigr) \\ &\quad\leqslant \bigl\Vert F(\theta) \bigr\Vert \bigl\Vert w^{(m)}(\theta) \bigr\Vert + L_{G}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}\bigl\Vert w^{(m)}(\theta) \bigr\Vert \\ &\quad\leqslant \frac{\delta_{1}}{2}\bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} + \frac{1}{2\delta_{1}}\bigl\Vert F(\theta) \bigr\Vert ^{2} + L_{G}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2}, \end{aligned}$$
where we also used assumption (A1), the Cauchy-Schwarz inequality, the Young inequality, and the facts
$$\begin{aligned} \bigl\Vert w^{(m)}(\theta) \bigr\Vert \leqslant \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}} \quad\mbox{and}\quad \bigl\Vert w^{(m)}(\theta) \bigr\Vert \leqslant\sup _{s\leqslant0}e^{\gamma s}\bigl\Vert w^{(m)}(\theta+s) \bigr\Vert =\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}. \end{aligned}$$
(3.4)
Consequently,
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\theta} \bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2} + \delta_{1}\bigl\Vert w^{(m)}(\theta ) \bigr\Vert _{\widehat{V}}^{2} \leqslant\frac{1}{\delta_{1}}\bigl\Vert F(\theta) \bigr\Vert ^{2} + 2L_{G}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2}. \end{aligned}$$
(3.5)
Integrating this inequality from τ to t, for any \(t\geqslant \tau\), we have
$$\begin{aligned} &\bigl\Vert w^{(m)}(t) \bigr\Vert ^{2} + \delta_{1} \int_{\tau}^{t} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \,{\mathrm{d}}\theta \\ &\quad\leqslant \bigl\Vert w^{(m)}(\tau) \bigr\Vert ^{2} + \frac{1}{\delta_{1}} \int_{\tau}^{t} \bigl\Vert F(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 2L_{G} \int_{\tau}^{t} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta. \end{aligned}$$
(3.6)
Multiplying (3.2) by \(\lambda_{j}v_{j}\), where \(\lambda_{j}\) is the eigenvalue associated with the eigenvector \(v_{j}\), and summing from \(j=1\) to m, we have
$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}\theta} \bigl\langle Aw^{(m)}(\theta), w^{(m)}(\theta) \bigr\rangle + \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2} + \bigl\langle B\bigl(u^{(m)},w^{(m)} \bigr), Aw^{(m)}(\theta) \bigr\rangle \\ & \qquad{} + \bigl\langle N \bigl(w^{(m)}(\theta)\bigr), Aw^{(m)}(\theta) \bigr\rangle \\ &\quad= \bigl(F(\theta), Aw^{(m)}(\theta)\bigr) + \bigl(G\bigl(\theta ,w_{\theta}^{(m)}\bigr), Aw^{(m)}(\theta)\bigr), \quad \theta\in(\tau,t]. \end{aligned}$$
(3.7)
On one hand, from (2.6), (2.8), Young’s inequality, and the facts \(\Vert \nabla u^{(m)} \Vert \leqslant \Vert \nabla w^{(m)} \Vert \) and \(\Vert u^{(m)} \Vert \leqslant \Vert w^{(m)} \Vert \) it follows that
$$\begin{aligned}& \bigl\vert \bigl\langle B\bigl(u^{(m)}(\theta), w^{(m)}(\theta)\bigr), Aw^{(m)}(\theta) \bigr\rangle \bigr\vert \\& \quad\leqslant \lambda\bigl\Vert u^{(m)} \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \nabla u^{(m)} \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \nabla w^{(m)} \bigr\Vert ^{\frac{1}{2}} \bigl\Vert Aw^{(m)} \bigr\Vert ^{\frac{1}{2}} \bigl\Vert Aw^{(m)} \bigr\Vert \\& \quad\leqslant\lambda\bigl\Vert w^{(m)} \bigr\Vert ^{\frac{1}{2}} \bigl\Vert w^{(m)} \bigr\Vert _{\widehat{V}} \bigl\Vert Aw^{(m)} \bigr\Vert ^{\frac{3}{2}} \\& \quad\leqslant \frac{1}{4}\bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2} + \frac{27\lambda^{4}}{4}\bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{4} \end{aligned}$$
(3.8)
and
$$\begin{aligned} \bigl\vert \bigl\langle N\bigl(w^{(m)}(\theta)\bigr), Aw^{(m)}(\theta) \bigr\rangle \bigr\vert \leqslant\frac{1}{4} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2} + c^{2}(\nu_{r})\bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2}. \end{aligned}$$
(3.9)
On the other hand,
$$\begin{aligned} \bigl(F(\theta),Aw^{(m)}(\theta)\bigr) \leqslant\bigl\Vert F(\theta) \bigr\Vert \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert \leqslant2\bigl\Vert F(\theta) \bigr\Vert ^{2} + \frac{1}{8}\bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2} \end{aligned}$$
(3.10)
and
$$\begin{aligned} \bigl(G\bigl(\theta,w_{\theta}^{(m)} \bigr),Aw^{(m)}(\theta)\bigr) &\leqslant\bigl\Vert G\bigl(\theta ,w_{\theta}^{(m)}\bigr) \bigr\Vert \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert \leqslant L_{G}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert \\ &\leqslant2L_{G}^{2}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} + \frac{1}{8}\bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2}. \end{aligned}$$
(3.11)
Substituting (3.8)-(3.11) into (3.7) and using (2.1), we get
$$\begin{aligned}& 2 \frac{\mathrm{d}}{\mathrm{d}\theta} \bigl\langle Aw^{(m)}(\theta ),w^{(m)}(\theta) \bigr\rangle + \frac{1}{2} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2} \\& \quad\leqslant \frac{27\lambda^{4}}{2}\bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2}\bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{4} \\& \quad \quad {}+ 2c^{2}(\nu_{r})\bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} + 4 \bigl\Vert F(\theta) \bigr\Vert ^{2} + 4L_{G}^{2} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \\& \quad\leqslant \biggl( \frac{27c_{2}\lambda^{4}}{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2}\bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} + 2c_{2}c^{2}(\nu _{r}) \biggr) \bigl\langle Aw^{(m)}(\theta ),w^{(m)}(\theta) \bigr\rangle \\& \quad\quad{} + 4\bigl\Vert F(\theta) \bigr\Vert ^{2} + 4L_{G}^{2} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2}. \end{aligned}$$
(3.12)
Set
$$\begin{aligned} &H_{m}(\theta) := 2 \bigl\langle Aw^{(m)}(\theta ),w^{(m)}(\theta) \bigr\rangle , \\ & I(\theta) := 4\bigl\Vert F( \theta) \bigr\Vert ^{2} + 4L_{G}^{2}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2}, \\ &K_{m}(\theta) := \frac{27c_{2}\lambda^{4}}{2}\bigl\Vert w^{(m)}( \theta) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} + 2c_{2}c^{2}(\nu _{r}). \end{aligned}$$
Then, (3.12) yields that
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\theta} H_{m}(\theta) \leqslant K_{m}(\theta)H_{m}(\theta) + I(\theta). \end{aligned}$$
(3.13)
Applying the Gronwall inequality to (3.13), for \(\tau\leqslant \tilde{r}\leqslant s\leqslant t\), we have
$$\begin{aligned} H_{m}(s) \leqslant\biggl(H_{m}(\tilde{r}) + \int_{\tau}^{t} I(\theta)\,\mathrm{d}\theta\biggr) \exp\biggl\{ \int_{\tau}^{t} K_{m}(\theta)\,\mathrm{d} \theta\biggr\} . \end{aligned}$$
(3.14)
Integrating this inequality for from τ to s, we obtain
$$\begin{aligned} (s-\tau)H_{m}(s) \leqslant\biggl( \int_{\tau}^{s} H_{m}(\tilde{r})\,\mathrm{d}r + (s-\tau) \int_{\tau}^{t} I(\theta)\,\mathrm{d}\theta\biggr) \exp\biggl\{ \int_{\tau}^{t} K_{m}(\theta)\,\mathrm{d} \theta\biggr\} . \end{aligned}$$
In particular, for any \(\tau+\epsilon\leqslant s\leqslant t\), \(n\geqslant1\), we have
$$\begin{aligned} H_{m}(s) \leqslant\biggl( \frac{1}{\epsilon} \int_{\tau}^{t} H_{m}(\tilde{r})\,\mathrm{d}r + \int_{\tau}^{t} I(\theta)\,\mathrm{d}\theta\biggr) \exp\biggl\{ \int_{\tau}^{t} K_{m}(\theta)\,\mathrm{d} \theta\biggr\} . \end{aligned}$$
(3.15)
By (3.6) we have
$$\begin{aligned} \int_{\tau}^{t} K_{m}(\theta)\,\mathrm{d} \theta&= \int_{\tau}^{t} \biggl( \frac{27c_{2}\lambda^{4}}{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} + 2c_{2}c^{2}(\nu_{r}) \biggr)\,\mathrm{d}\theta \\ &\leqslant \frac{27c_{2}\lambda^{4}}{2} \sup_{\theta\in[\tau ,t]}\bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2} \int_{\tau}^{t} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \,{\mathrm{d}}\theta+ 2c_{2}c^{2}(\nu_{r}) (t-\tau) \\ &\leqslant \frac{27c_{2}\lambda^{4}}{2\delta_{1}} \biggl( \bigl\Vert w^{(m)}(\tau) \bigr\Vert ^{2} + \frac{1}{\delta_{1}} \int_{\tau}^{t} \bigl\Vert F(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 2L_{G} \int_{\tau}^{t} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta \biggr)^{2} \\ & \quad{}+ 2c_{2}c^{2}(\nu_{r}) (t- \tau). \end{aligned}$$
(3.16)
From (2.1) and (3.6) it follows that
$$\begin{aligned} \int_{\tau}^{t} H_{m}(\tilde{r})\,\mathrm{d} \tilde{r} &= 2\int_{\tau}^{t} \bigl\langle Aw^{(m)}(\tilde {r}),w^{(m)}(\tilde{r}) \bigr\rangle \,{\mathrm{d}}\tilde{r} \leqslant2 c_{1}^{-1} \int_{\tau}^{t} \bigl\Vert w^{(m)}( \tilde{r}) \bigr\Vert _{\widehat{V}}^{2} \,{\mathrm{d}}\tilde{r} \\ &\leqslant2 c_{1}^{-1}\delta_{1}^{-1} \bigl\Vert w^{(m)}(\tau) \bigr\Vert ^{2} + 2 c_{1}^{-1}\delta_{1}^{-2} \int_{\tau}^{t} \bigl\Vert F(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta \\ &\quad {}+ 4 c_{1}^{-1}\delta _{1}^{-1}L_{G} \int_{\tau}^{t} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta. \end{aligned}$$
(3.17)
In addition,
$$\begin{aligned} (t-\tau) \int_{\tau}^{t} I(\theta)\,\mathrm{d}\theta= 4(t-\tau ) \int_{\tau}^{t} \bigl\Vert F(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 4L_{G}^{2}(t-\tau) \int_{\tau}^{t} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta. \end{aligned}$$
(3.18)
Similarly to (2.11), we have
$$\begin{aligned} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \leqslant e^{-(\delta_{1}\lambda_{1}-2L_{G})(\theta -\tau)}\bigl\Vert \phi(s) \bigr\Vert _{\gamma}^{2} + \frac{2}{\delta_{1}} \int_{\tau}^{\theta} e^{-(\delta_{1}\lambda_{1}-2L_{G})(\theta -r)} \bigl\Vert F(r) \bigr\Vert ^{2}\,\mathrm{d}r. \end{aligned}$$
(3.19)
Thus,
$$\begin{aligned} \int_{\tau}^{t} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta&\leqslant \frac{2}{\delta_{1}(\delta_{1}\lambda_{1}-2L_{G})} \int_{\tau}^{t} \bigl[e^{(\delta_{1}\lambda_{1}-2L_{G})(r-\tau )}-e^{(\delta_{1}\lambda_{1}-2L_{G})(r-t)} \bigr] \bigl\Vert F(r) \bigr\Vert ^{2} \,{\mathrm{d}}r \\ &\quad {}+ \frac{1-e^{-(\delta_{1}\lambda_{1}-2L_{G})(t-\tau)}}{\delta _{1}\lambda_{1}-2L_{G}} \bigl\Vert \phi(s) \bigr\Vert _{\gamma}^{2}. \end{aligned}$$
(3.20)
Taking (2.1), (3.15)-(3.18), and (3.20) into account, we complete the proof of assertion (i).
Now integrating (3.12) for θ between \(\tau+\epsilon\) and t, we get
$$\begin{aligned} & \int_{\tau+\epsilon}^{t} \bigl\Vert Aw^{(m)}(\theta ) \bigr\Vert ^{2} \,{\mathrm{d}}\theta \\ &\quad \leqslant 2 c_{1}^{-1} \bigl\Vert w^{(m)}(\tau +\epsilon) \bigr\Vert _{\widehat{V}}^{2} + 27\lambda ^{4} \int_{\tau+\epsilon}^{t} \bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{4} \,{\mathrm{d}}\theta \\ &\quad \quad {}+ 4c^{2}(\nu_{r}) \int_{\tau+\epsilon}^{t} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \,{\mathrm{d}}\theta+ 8 \int_{\tau+\epsilon}^{t} \bigl\Vert F(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 8L_{G}^{2} \int_{\tau+\epsilon}^{t} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2}\,{\mathrm{d}}\theta, \end{aligned}$$
which, together with assertion (i), (3.6), and (3.20), implies assertion (ii).
Finally, multiplying (3.2) by \(\beta_{m,j}'(t)\), summing them from \(j=1\) to n, and replacing the variable t with θ, we obtain
$$\begin{aligned} &\bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} + \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\theta} \bigl\langle Aw^{(m)}(\theta), w^{(m)}(\theta) \bigr\rangle + \bigl\langle B\bigl(u^{(m)}(\theta),w^{(m)}(\theta)\bigr),w^{(m)'}( \theta) \bigr\rangle \\ & \qquad{}+ \bigl\langle N \bigl(w^{(m)}(\theta)\bigr), w^{(m)'}(\theta) \bigr\rangle \\ &\quad = \bigl(F(\theta,x), w^{(m)'}(\theta)\bigr) + \bigl(G\bigl(\theta ,w_{\theta}^{(m)}\bigr), w^{(m)'}(\theta)\bigr). \end{aligned}$$
(3.21)
Observe that
$$\begin{aligned}& \begin{aligned}[b] & \bigl(F(\theta,x), w^{(m)'}(\theta)\bigr) + \bigl(G \bigl(\theta,w_{\theta}^{(m)}\bigr), w^{(m)'}(\theta) \bigr) \\ &\quad \leqslant \bigl(\bigl\Vert F(\theta) \bigr\Vert + \bigl\Vert G\bigl( \theta,w_{\theta}^{(m)}\bigr) \bigr\Vert \bigr) \bigl\Vert w^{(m)'}(\theta) \bigr\Vert \\ &\quad \leqslant2\bigl\Vert F(\theta) \bigr\Vert ^{2} + 2L_{G}^{2}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} + \frac{1}{4} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2}. \end{aligned} \end{aligned}$$
(3.22)
By Lemma 2.1 we deduce that
$$\begin{aligned}& \bigl\vert \bigl\langle B\bigl(u^{(m)}(\theta ),w^{(m)}(\theta)\bigr), w^{(m)'}(\theta) \bigr\rangle \bigr\vert \\& \quad \leqslant \lambda\bigl\Vert u^{(m)}(\theta) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \nabla u^{(m)}(\theta) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \nabla w^{(m)}(\theta) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert \\& \quad \leqslant \lambda\bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert \\& \quad \leqslant \lambda^{2}\bigl\Vert w^{(m)}(\theta) \bigr\Vert \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert + \frac{1}{4} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} \end{aligned}$$
(3.23)
and
$$\begin{aligned} \bigl\vert \bigl\langle N\bigl(w^{(m)}(\theta)\bigr), w^{(m)'}(\theta) \bigr\rangle \bigr\vert \leqslant c^{2}(\nu _{r}) \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} + \frac{1}{4} \bigl\Vert w^{(m)'}( \theta) \bigr\Vert ^{2}. \end{aligned}$$
(3.24)
It follows from (3.21)-(3.24) that
$$\begin{aligned}& \bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} + \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\theta} \bigl\langle Aw^{(m)}(\theta ),w^{(m)}(\theta) \bigr\rangle \\& \quad = - \bigl\langle B\bigl(u^{(m)}(\theta),w^{(m)}(\theta) \bigr), w^{(m)'}(\theta) \bigr\rangle - \bigl\langle N \bigl(w^{(m)}(\theta)\bigr), w^{(m)'}(\theta) \bigr\rangle \\& \quad \quad {} + \bigl(F(\theta,x), w^{(m)'}(\theta)\bigr) + \bigl(G\bigl(\theta ,w_{\theta}^{(m)}\bigr), w^{(m)'}(\theta)\bigr) \\& \quad \leqslant \frac{3}{4}\bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} + 2\bigl\Vert F(\theta) \bigr\Vert ^{2} + 2L_{G}^{2}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} + \lambda^{2}\bigl\Vert w^{(m)}(\theta) \bigr\Vert \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert \\& \quad \quad {}+ c^{2}(\nu_{r}) \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2}. \end{aligned}$$
Thus,
$$\begin{aligned} &\bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} + 2 \frac{\mathrm{d}}{\mathrm{d}\theta} \bigl\langle Aw^{(m)}(\theta ),w^{(m)}(\theta )\bigr\rangle \\ &\quad \leqslant 8\bigl\Vert F(\theta) \bigr\Vert ^{2} + 8L_{G}^{2}\bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} + 4\lambda^{2}\bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{3} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert + 4c^{2}(\nu_{r}) \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2}. \end{aligned}$$
Integrating this inequality and using (2.1), we get that
$$\begin{aligned} \int_{\tau+\epsilon}^{t} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta &\leqslant 2c_{1}^{-1} \bigl\Vert w^{(m)}(\tau+\epsilon) \bigr\Vert ^{2} + 8 \int_{\tau+\epsilon}^{t} \bigl\Vert F(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 8L_{G}^{2} \int_{\tau+\epsilon}^{t} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta \\ &\quad {}+ 2\lambda^{2}\sup_{\theta\in[\tau+\epsilon,t]} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \int_{\tau+\epsilon}^{t} \bigl(\bigl\Vert w^{(m)}( \theta) \bigr\Vert _{\widehat{V}}^{2} + \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2}\bigr)\,\mathrm{d}\theta \\ &\quad {}+ 4c^{2}(\nu_{r}) \int_{\tau+\epsilon}^{t} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \,{\mathrm{d}}\theta, \end{aligned}$$
which, together with (3.6), (3.20), and assertions (i)-(ii), gives assertion (iii). The proof is complete. □

Corollary 3.1

Under the conditions of Lemma  3.1, for any bounded set \(\mathcal {B}\subset\mathcal{C}_{\gamma}(\widehat{H})\) and any \(\tau\in\mathbb {R}, \epsilon>0, t\geqslant\tau+\epsilon\), the set \(\bigcup_{s\in [\tau+\epsilon, t]}U(s,\tau)\mathcal{B}\) is bounded in .

Proof

In [25], the authors proved that, for any \(w_{\tau}=\phi(s)\in \mathcal{C}_{\gamma}(\widehat{H})\), the Galerkin approximation solutions \(\{w^{(m)}(\cdot; \tau,w_{\tau})\}_{m\geqslant1}\) converge weakly to \(w(\cdot; \tau,w_{\tau})\) in \(L^{2}(\tau ,t;\widehat{V})\) and \(w(\cdot; \tau,w_{\tau})\in\mathcal{C}([\tau ,t];\widehat{H})\). So Corollary 3.1 is a straightforward consequence of Lemma 2.4 and Lemma 3.1(i). □

By increasing the regularity of \(F(t,x)\) and \(G(t,w_{t})\) properly we can improve the results of Lemma 3.1.

Lemma 3.2

Assume that (A1) and (A2) hold. Then, for any bounded set \(\mathcal{B}\subset\mathcal{C}_{\gamma }(\widehat{H})\) and any \(\tau\in\mathbb{R}, \epsilon>0, t \geqslant \tau+\epsilon\), the following properties are fulfilled:
  1. (iv)

    the set \(\{w^{(m)'}(s;\tau,w_{\tau}) \mid s\in[\tau+2\epsilon, t], w_{\tau}=\phi(s)\in\mathcal{B}\}\) is bounded in Ĥ;

     
  2. (v)

    the set \(\{w^{(m)}(s;\tau,w_{\tau}) \mid s\in[\tau +2\epsilon, t], w_{\tau}=\phi(s)\in\mathcal{B}\}\) is bounded in \(D(A)=\widehat{V}\cap(H^{2})^{3}\).

     

Proof

Without loss of generality, we consider a fixed bounded set \(\mathcal {B}\subset\mathcal{C}_{\gamma}(\widehat{H})\). Differentiating equation (3.2) with respect to time and multiplying the resulting equation by \(\beta_{m,j}'(t)\) and summing them from \(j=1\) to m, we have
$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl\Vert w^{(m)'}(t) \bigr\Vert ^{2} + \bigl\langle Aw^{(m)'}(t), w^{(m)'}(t) \bigr\rangle + \bigl\langle \bigl( B \bigl(u^{(m)}(t),w^{(m)}(t)\bigr) \bigr)', w^{(m)'}(t) \bigr\rangle \\ & \qquad{} + \bigl(N \bigl(w^{(m)'}(t)\bigr), w^{(m)'}(t)\bigr) \\ &\quad= \bigl(F'(t), w^{(m)'}(t)\bigr) + \bigl( \bigl(G(t,w_{t})\bigr)', w^{(m)'}(t)\bigr). \end{aligned}$$
(3.25)
In the following, we make a more detailed estimate for each term in (3.25). First, from Lemma 2.1 and the Cauchy-Schwarz inequality it is easy to see that
$$\begin{aligned} \delta_{1} \bigl\Vert w^{(m)'}(t) \bigr\Vert _{\widehat{V}}^{2} \leqslant\bigl\langle Aw^{(m)'}(t), w^{(m)'}(t) \bigr\rangle + \bigl\langle N\bigl(w^{(m)'}(t)\bigr), w^{(m)'}(t) \bigr\rangle \end{aligned}$$
(3.26)
and
$$\begin{aligned} &\bigl\vert \bigl\langle \bigl(B\bigl(u^{(m)}(t),w^{(m)}(t) \bigr)\bigr)', w^{(m)'}(t) \bigr\rangle \bigr\vert \\ &\quad = \bigl\vert \bigl\langle B\bigl(u^{(m)'}(t),w^{(m)}(t)\bigr), w^{(m)'}(t) \bigr\rangle \bigr\vert \\ &\quad\leqslant \lambda\bigl\Vert u^{(m)'}(t) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \nabla u^{(m)'}(t) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert w^{(m)'}(t) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \nabla w^{(m)'}(t) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \nabla w^{(m)}(t) \bigr\Vert \\ &\quad\leqslant \lambda\bigl\Vert w^{(m)'}(t) \bigr\Vert \bigl\Vert w^{(m)}(t) \bigr\Vert _{\widehat{V}} \bigl\Vert w^{(m)'}(t) \bigr\Vert _{\widehat{V}} \\ &\quad\leqslant \delta_{1}^{-1}\lambda^{2} \bigl\Vert w^{(m)'}(t) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(t) \bigr\Vert _{\widehat{V}}^{2} + \frac{\delta_{1}}{4} \bigl\Vert w^{(m)'}(t) \bigr\Vert _{\widehat{V}}^{2}. \end{aligned}$$
(3.27)
Then, under assumption (A2)(I) and (3.4), we have
$$\begin{aligned} &\bigl(F'(t), w^{(m)'}(t)\bigr) + \bigl( \bigl(G(t,w_{t})\bigr)', w^{(m)'}(t)\bigr) \\ &\quad\leqslant 2\delta_{1}^{-1} \bigl\Vert F'(t) \bigr\Vert ^{2} + 2\delta_{1}^{-1} \tilde{L}_{G}^{2} \bigl\Vert w_{t}^{(m)'} \bigr\Vert _{\gamma}^{2} + \frac{\delta_{1}}{4}\bigl\Vert w^{(m)'}(t) \bigr\Vert _{\widehat{V}}^{2}. \end{aligned}$$
(3.28)
Now, taking (3.25)-(3.28) into account, we obtain
$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \bigl\Vert w^{(m)'}(t) \bigr\Vert ^{2} + \delta_{1} \bigl\Vert w^{(m)'}(t) \bigr\Vert _{\widehat{V}}^{2} \\ &\quad\leqslant 2\delta_{1}^{-1}\lambda^{2} \bigl\Vert w^{(m)'}(t) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(t) \bigr\Vert _{\widehat{V}}^{2} + 4\delta _{1}^{-1} \bigl\Vert F'(t) \bigr\Vert ^{2} + 4\delta_{1}^{-1}\tilde{L}_{G}^{2} \bigl\Vert w_{t}^{(m)'} \bigr\Vert _{\gamma}^{2}. \end{aligned}$$
Replacing the variable t with θ and integrating it between r and s, we see that, for all \(\tau\leqslant r\leqslant s\leqslant t\),
$$\begin{aligned} &\bigl\Vert w^{(m)'}(s) \bigr\Vert ^{2} +\delta_{1} \int_{r}^{s} \bigl\Vert w^{(m)'}(\theta ) \bigr\Vert _{\widehat{V}}^{2}\,{\mathrm{d}}\theta \\ &\quad\leqslant \bigl\Vert w^{(m)'}(r) \bigr\Vert ^{2} + 2 \delta_{1}^{-1}\lambda^{2} \int_{r}^{t} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \,{\mathrm{d}}\theta \\ & \qquad{}+ 4\delta_{1}^{-1}\int_{r}^{t} \bigl\Vert F'(\theta)\bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 4\delta_{1}^{-1}\tilde{L}_{G}^{2} \int_{r}^{t} \bigl\Vert w_{\theta}^{(m)'}\bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta. \end{aligned}$$
(3.29)
Particularly, for all \(\tau+2\epsilon\leqslant r+\epsilon\leqslant s \leqslant t\), we have
$$\begin{aligned} \bigl\Vert w^{(m)'}(s) \bigr\Vert ^{2} &\leqslant \bigl\Vert w^{(m)'}(r) \bigr\Vert ^{2} + 2\delta _{1}^{-1}\lambda^{2}\sup_{\theta\in[\tau+\epsilon,t]} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \int_{\tau+\epsilon}^{t} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta \\ &\quad {}+ 4\delta_{1}^{-1} \int_{\tau+\epsilon}^{t} \bigl\Vert F'(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 4\delta_{1}^{-1} \tilde{L}_{G}^{2} \int_{\tau+\epsilon}^{t} \bigl\Vert w_{\theta}^{(m)'} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta. \end{aligned}$$
(3.30)
Integrating this inequality with respect to r between \(\tau+\epsilon \) and s, we have
$$\begin{aligned} \bigl\Vert w^{(m)'}(s) \bigr\Vert ^{2} &\leqslant \frac{1}{s-\tau-\epsilon} \int_{\tau+\epsilon}^{t} \bigl\Vert w^{(m)'}(r) \bigr\Vert ^{2} \,{\mathrm{d}}r \\ &\quad {}+ \frac{2\lambda^{2}}{\delta_{1}}\sup _{\theta\in[\tau+\epsilon,t]} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} \int_{\tau+\epsilon}^{t} \bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta \\ &\quad {}+ 4\delta_{1}^{-1} \int_{\tau+\epsilon}^{t} \bigl\Vert F'(\theta) \bigr\Vert ^{2} \,{\mathrm{d}}\theta+ 4\delta_{1}^{-1} \tilde{L}_{G}^{2} \int_{\tau+\epsilon}^{t} \bigl\Vert w_{\theta}^{(m)'} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta \end{aligned}$$
(3.31)
for all \(\tau+2\epsilon\leqslant r+\epsilon\leqslant s \leqslant t\). Then, it is not difficult to get that, by using the same proof as (3.21) in [25],
$$\begin{aligned} \bigl\Vert w_{\theta}^{(m)'} \bigr\Vert _{\gamma}^{2} \leqslant e^{-\delta_{1}\lambda_{1}(\theta-\tau )}\bigl\Vert \phi '(s) \bigr\Vert _{\gamma}^{2} + \frac{2}{\delta_{1}} \int_{\tau}^{\theta} e^{-(\delta_{1}\lambda_{1}-2\tilde {L}_{G})(\theta-r)} \bigl\Vert F'(r) \bigr\Vert ^{2} \,{\mathrm{d}}r. \end{aligned}$$
(3.32)
Thus, we deduce that
$$\begin{aligned} \int_{\tau}^{t} \bigl\Vert w_{\theta}^{(m)'} \bigr\Vert _{\gamma}^{2} \,{\mathrm{d}}\theta&\leqslant \frac{2}{\delta_{1}(\delta_{1}\lambda_{1}-2\tilde{L}_{G})} \int_{\tau}^{t} \bigl[e^{(\delta_{1}\lambda_{1}-2\tilde {L}_{G})(r-\tau)} -e^{(\delta_{1}\lambda_{1}-2\tilde{L}_{G})(r-t)}\bigr] \bigl\Vert F'(r) \bigr\Vert ^{2} \,{\mathrm{d}}r \\ &\quad {}+ \frac{1-e^{-\delta_{1}\lambda_{1}(t-\tau)}}{\delta_{1}\lambda _{1}} \bigl\Vert \phi'(s) \bigr\Vert _{\gamma}^{2}, \end{aligned}$$
(3.33)
which, combined with assumption (A2)(II), yields the boundedness of \(\int_{\tau}^{t} \Vert w_{\theta}^{(m)'} \Vert _{\gamma}^{2} \,{\mathrm{d}}\theta\). Consequently, property (iv) follows from (3.31), (3.33), assumption (A2), and Lemma 3.1.
Next, we prove property (v). Multiplying (3.2) by \(\lambda _{j}\beta_{m,j}(t)\) and summing the resulting equation from \(j=1\) to m, we obtain
$$\begin{aligned} &\bigl(w^{(m)'}(\theta),Aw^{(m)}(\theta)\bigr) + \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2} + \bigl\langle B\bigl(u^{(m)},w^{(m)}\bigr), Aw^{(m)}(\theta) \bigr\rangle + \bigl\langle N\bigl(w^{(m)}\bigr), Aw^{(m)}( \theta) \bigr\rangle \\ &\quad = \bigl(F(\theta), Aw^{(m)}(\theta)\bigr) + \bigl(G\bigl(\theta ,w_{\theta}^{(m)}\bigr), Aw^{(m)}(\theta)\bigr). \end{aligned}$$
(3.34)
Observe that
$$\begin{aligned} \bigl(w^{(m)'}(\theta), Aw^{(m)}(\theta)\bigr) \leqslant2\bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} + \frac{1}{8}\bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2}, \end{aligned}$$
(3.35)
which, together with (3.8)-(3.11) and (3.34), gives that, for any \(\theta>\tau\),
$$\begin{aligned} \bigl\Vert Aw^{(m)}(\theta) \bigr\Vert ^{2} &\leqslant 54\lambda^{4}\bigl\Vert w^{(m)}(\theta) \bigr\Vert ^{2} \bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{4} + 8c^{2}(\nu_{r})\bigl\Vert w^{(m)}(\theta) \bigr\Vert _{\widehat{V}}^{2} + 16\bigl\Vert w^{(m)'}(\theta) \bigr\Vert ^{2} \\ &\quad {}+ 16 \bigl\Vert F(\theta) \bigr\Vert ^{2} + 16 L_{G}^{2} \bigl\Vert w_{\theta}^{(m)} \bigr\Vert _{\gamma}^{2}. \end{aligned}$$
(3.36)
Since \(W_{\mathrm{loc}}^{1,2}(\mathbb{R};\widehat{H}) \hookrightarrow\mathcal{C}(\mathbb{R};\widehat{H})\) and \(f\in W_{\mathrm{loc}}^{1,2}(\mathbb{R};\widehat{H})\), so \(f\in\mathcal{C}(\mathbb{R};\widehat{H})\), which, together with (3.19), (3.36), Lemmas 3.1(i) and 3.2(iv), implies property (v). The proof is complete. □

At this state, we give the proof of the main results of this paper.

Proof of Theorem 1.1

(1) According to Lemma 3.1, following the standard diagonal procedure, there exists a function \(w(\cdot)\) such that (by extracting a subsequence if necessary)
$$\begin{aligned} w^{(m)}(\cdot) & \rightharpoonup^{*} w(\cdot) \mbox{ weakly star in } L^{\infty}(\tau+\epsilon, t; \widehat{V}), \end{aligned}$$
(3.37)
$$\begin{aligned} w^{(m)}(\cdot) & \rightharpoonup w(\cdot) \mbox{ weakly in } L^{2}\bigl(\tau+\epsilon, t; D(A)\bigr), \end{aligned}$$
(3.38)
$$\begin{aligned} w^{(m)'}(\cdot) & \rightharpoonup w'(\cdot) \mbox{ weakly in } L^{2}(\tau+\epsilon, t; \widehat{H}). \end{aligned}$$
(3.39)
Furthermore, \(w(\cdot)\in\mathcal{C}([t+\epsilon, t]; \widehat{V})\). It follows from the uniqueness of the limit function that \(w(\cdot)\) is a weak solution of system (2.10). Then, part (1) of Theorem 1.1 is a consequence of Lemmas 2.4 and 3.2.
(2) It is not difficult to see that if \(\tau< T_{1}-1\) is fixed, then
$$\begin{aligned} \bigcup_{t\in[T_{1},T_{2}]} \mathcal{A}(t) \subset\bigcup _{t\in[\tau+1, T_{2}]} U(t,\tau) \mathcal{A}(\tau). \end{aligned}$$
Consequently, combining Lemma 3.2 and part (1), we obtain the boundedness result of part (2). The proof is complete. □

Declarations

Acknowledgements

The authors sincerely thank Professor Yeping Li and Associate Professor Jie Liao of East China University of Science and Technology for their many useful suggestions and elaborate guidance. The authors also warmly thank the anonymous reviewers for their pertinent comments and suggestions, which greatly improved the earlier manuscript. Wenlong Sun is supported by the NSFC of China (No. 11671134) and China Scholarship Council.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Shanghai Dianji University
(2)
School of Mathematical Sciences, Shanghai Jiao Tong University
(3)
Department of Mathematics, East China University of Science and Technology

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