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On global dynamics of 2D convective Cahn–Hilliard equation

Abstract

In this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in \(H^{4}(\Omega )\) when the initial value belongs to \(H^{1}(\Omega )\).

Introduction

The dynamic properties of diffusion equations ensure the stability of diffusion phenomena and provide the mathematical foundation for the study of diffusion dynamics. There are many studies on the existence of global attractors for diffusion equations. For the classical results, we refer the reader to [19].

The convective Cahn–Hilliard equation [1016], which arises naturally as a continuous model for the formation of facets and corners in crystal growth, is a typical fourth order nonlinear parabolic equation. Let \(\Omega =[0,L]\times [0,L]\), where \(L>0\), γ is a positive constant, β⃗ is a vector. We consider the convective Cahn–Hilliard equation in the 2D case:

$$ u_{t}+\gamma \Delta ^{2}u=\Delta \varphi (u)- \vec{\beta }\cdot \nabla \psi (u),\quad x=(x_{1},x_{2})\in \mathbb{R}^{2},t\geq 0. $$
(1)

Equation (1) is supplemented by the following boundary conditions:

$$ u(x_{1}+L,x_{2},t)=u(x_{1},x_{2}+L,t)=u(x_{1},x_{2},t), \quad x\in \mathbb{R}^{2},t\geq 0, $$
(2)

and the initial condition

$$ u(x,0)=u_{0}(x). $$
(3)

In this paper, we denote by \(H=L^{2}(\Omega )\), \((\cdot ,\cdot )\) the H-inner product and by \(\|\cdot \|\) the corresponding H-norm, denote \(A=-\Delta \), where Δ is the Laplace operator. Assume that the initial function has zero mean, i.e., \(\int _{\Omega }u_{0}(x)\,dx=0\), then it follows that \(\int _{\Omega }u(x,t)\,dx=0\) for \(t>0\). Here, as [3], we set

$$ \dot{H}^{k}_{per}= \biggl\{ u\Big|u\in H^{k}_{per}( \Omega ), \int _{\Omega }u(x,t)\,dx=0, \biggr\} , \quad k=1,2,\ldots . $$

Using the same method as [13], we obtain the lemma on the existence of global weak solution to problem (1)–(3).

Lemma 1.1

Suppose that \(u_{0}\in \dot{H}_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{2}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then there exists a unique solution u for problem (1)(3) such that

$$ u\in C \bigl(\mathbb{R}^{+};\dot{H}_{per}^{1}( \Omega ) \bigr) \cap L_{loc}^{2} \bigl( \mathbb{R}^{+};\dot{H}_{per}^{2}(\Omega ) \bigr). $$

By Lemma 1.1, we can define the operator semigroup \(S(t)u_{0}:\dot{H}^{1}_{per}(\Omega )\times \mathbb{R}^{+} \rightarrow \dot{H}^{1}_{per}(\Omega ) \), which is \((\dot{H}^{1}_{per},\dot{H}^{1}_{per})\)-continuous. In what follows, we always assume that \(\{S(t)\}_{t\geq 0}\) is the semigroup generated by the weak solutions of problem (1). It is sufficient to see that the restriction of \(\{S(t)\}\) on the affined space \(\dot{H}^{1}_{per}(\Omega )\) is a well-defined semigroup.

Proposition 1.2

([1719])

Suppose that \(\mathcal{A}\) is an \((H^{1}, H^{1})\)-global attractor for \(\{S(t)\}_{t\geq 0}\). Suppose further that \(\{S(t)\}_{t\geq 0}\) has a bounded \((H^{1}, H^{4})\)-absorbing set and \(\{S(t)\}_{t\geq 0}\) is \((H^{1}, H^{4})\)-asymptotically compact. Then \(\mathcal{A}\) is also an \((H^{1}, H^{4})\)-global attractor.

The main result of this paper will be stated in the following.

Theorem 1.3

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$
(4)

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then there exists an \((H^{1},H^{4})\)-global attractor for the solution \(u(x,t)\) of problem (1)(3), which is invariant and compact in \(H^{4}(\Omega )\) and attracts every bounded subset of \(H^{1}(\Omega )\) with respect to the norm topology of \(H^{4}(\Omega )\).

Remark 1.4

In the previous papers [18, 20, 21], my cooperators and I also studied the existence of global attractor for a 2D convective Cahn–Hilliard equation. There are two main differences between the previous results and Theorem 1.3. First, in [18, 20], we assumed that there exists double-well potential for the convective Cahn–Hilliard equation, which was replaced by the higher order polynomial in [21]. But, in this paper, this assumption is changed by (4), which seems more abroad than double-well potential and polynomial. Second, in [18], the existence of \((H^{2},H^{2})\)-global attractor was obtained, and in [20, 21], the existence of \((H^{k},H^{k})\)-global attractor was proved. In this paper, we only assume that the initial data belongs to \(H^{1}(\Omega )\) and obtain the \((H^{1},H^{4})\)-global attractor for the 2D convective Cahn–Hilliard equation.

The remaining parts are organized as follows. We begin by giving some uniform estimates of solutions for the 2D convective Cahn–Hilliard equation in Sect. 2. Then, in Sect. 3, we prove the main results on the existence of global attractor.

Uniform estimates of solutions

First of all, we establish the uniform estimates of solutions of problem (1) as \(t\rightarrow \infty \). These estimates are necessary to prove the existence of global attractors.

Lemma 2.1

Suppose that \(u_{0}\in L^{2}(\Omega )\) and the functions \(\varphi (r)\in C^{1}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \psi '(r)\leq c_{0}r \sqrt{\varphi '(r)}+c_{1}. $$

Then, for problem (1)(3), we have

$$ \bigl\Vert u(t) \bigr\Vert \leq M_{0}, \quad \forall t\geq T_{0}, $$

and

$$ \int _{t}^{t+1} \bigl\Vert A u(t) \bigr\Vert ^{2}\,d\tau \leq M_{0}, \quad t\geq T_{0}. $$

Here, \(M_{0}\) is a positive constant depending on γ and \(c_{i}\) \((i=0,1)\). \(T_{0}\) depends on γ, \(c_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|^{2}\leq R^{2}\).

Proof

Multiplying equation (1) by u and integrating the resulting relation over Ω, we obtain

$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert u \Vert ^{2}+\gamma \Vert \Delta u \Vert ^{2}+ \int _{\Omega } \varphi '(u) \vert \nabla u \vert ^{2}\,dx=\beta \cdot \int _{\Omega }\psi '(u)u \nabla u\,dx. \end{aligned} \end{aligned}$$
(5)

Note that

$$ \begin{aligned} \beta \cdot \int _{\Omega }\psi '(u)u\nabla u\,dx&=\beta \cdot \int _{ \Omega }\psi '(u)u\nabla u\,dx \\ &\leq c_{2} \vert \beta \vert \int _{\Omega } \bigl\vert u\nabla u\sqrt{\varphi '(u)} \bigr\vert \,dx+c_{3} \vert \beta \vert \int _{\Omega } \vert u \vert \,dx \\ &\leq \frac{1}{2} \int _{\Omega }\varphi '(u) \vert \nabla u \vert ^{2}\,dx+\frac{c_{2}}{2} \Vert u \Vert ^{2}+ \frac{c_{3}}{2}. \end{aligned} $$

Hence

$$ \frac{d}{dt} \Vert u \Vert ^{2}+2\gamma \Vert \Delta u \Vert ^{2}+ \int _{\Omega }\varphi '(u) \vert \nabla u \vert ^{2}\,dx\leq c_{2} \Vert u \Vert ^{2}+c_{3}. $$
(6)

Applying Poincaré’s inequality, we arrive at

$$ \Vert u \Vert ^{2}\leq c' \Vert \nabla u \Vert ^{2}. $$

Moreover,

$$ c' \Vert \nabla u \Vert ^{2}=-c' \int _{\Omega }u\Delta u\,dx\leq \frac{1}{2} \Vert u \Vert ^{2}+ \frac{(c')^{2}}{2} \Vert \Delta u \Vert ^{2}. $$

Therefore, the following inequality holds:

$$ \Vert u \Vert ^{2}\leq \bigl(c'\bigr)^{2} \Vert \Delta u \Vert ^{2}. $$

Summing up, we get

$$ \frac{d}{dt} \Vert u \Vert ^{2}+ \biggl( \frac{2\gamma }{(c')^{2}}-c_{4} \biggr) \Vert u \Vert ^{2}\leq c_{5}, $$
(7)

where γ satisfies \(\frac{2\gamma }{(c')^{2}}-c_{4}>0\). Using Gronwall’s inequality, we deduce that

$$ \Vert u \Vert ^{2}\leq e^{- (\frac{2\gamma }{(c')^{2}}-c_{2} )t} \Vert u_{0} \Vert ^{2}+\frac{c_{3}(c')^{2}}{2\gamma -c_{2}(c')^{2}}\leq \frac{2c_{3}(c')^{2}}{2\gamma -c_{2}(c')^{2}} $$
(8)

for all \(t\geq T^{*}=\frac{(c')^{2}}{2\gamma -c_{2}(c')^{2}}\ln \frac{[2\gamma -c_{2}(c')^{2}]R^{2}}{c_{3}(c')^{2}}\). Integrating (6) over \((t,t+1)\) with \(t\geq T^{*}\) yields

$$\begin{aligned} \int _{t}^{t+1} \Vert \Delta u \Vert ^{2}\,d\tau \leq c_{4}. \end{aligned}$$
(9)

By using a mean value theorem for integrals, we obtain the existence of a time \(t_{0}'\in (T^{*},T^{*}+1)\) such that

$$ \bigl\Vert \Delta u\bigl(t_{0}'\bigr) \bigr\Vert ^{2}\leq c_{5} $$

holds uniformly, the proof is complete. □

Lemma 2.2

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{2}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)(3), we have

$$ \bigl\Vert \nabla u(t) \bigr\Vert \leq M_{1}, \quad \forall t\geq T_{1}, $$

and

$$ \int _{t}^{t+1} \bigl\Vert \nabla \Delta u(t) \bigr\Vert ^{2}\,d\tau \leq M_{1}, \quad t\geq T_{1}. $$

Here, \(M_{1}\) is a positive constant depending on γ and \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{1}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).

Proof

Multiplying equation (1) by \(-\Delta u\) and integrating the resulting relation over Ω yields

$$ \begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \nabla u \Vert ^{2}+\gamma \Vert \nabla \Delta u \Vert ^{2} &=- \int _{\Omega }\Delta \varphi (u)\Delta u\,dx-\beta \cdot \int _{\Omega } \nabla \psi (u)\Delta u\,dx \\ &=- \int _{\Omega }\varphi '(u) \vert \Delta u \vert ^{2}\,dx- \int _{\Omega }\varphi {''}(u) \vert \nabla u \vert ^{2}\Delta u\,dx \\ &\quad{}-\beta \cdot \int _{\Omega }\psi '(u)\nabla u\Delta u\,dx. \end{aligned} $$

Hence

$$ \begin{aligned} & \frac{1}{2}\frac{d}{dt} \Vert \nabla u \Vert ^{2}+\gamma \Vert \nabla \Delta u \Vert ^{2}+ \int _{\Omega }\varphi '(u) \vert \Delta u \vert ^{2}\,dx \\ &\quad =- \int _{\Omega }\varphi {''}(u) \vert \nabla u \vert ^{2}\Delta u\,dx-\beta \cdot \int _{\Omega }\psi '(u)\nabla u\Delta u\,dx \\ &\quad \leq c \int _{\Omega } \vert u\Delta u \vert \vert \nabla u \vert ^{2}\,dx+c \vert \beta \vert \int _{ \Omega } \bigl\vert u^{2}\sqrt{\varphi '(u)}\nabla u\Delta u \bigr\vert \,dx+c \Vert \nabla u \Vert ^{2} \\ &\quad \leq \frac{c}{2} \int _{\Omega } \vert \nabla u \vert ^{4}\,dx+ \frac{c}{2} \int _{\Omega } \vert u \Delta u \vert ^{2}\,dx+ \int _{\Omega }\varphi '(u) \vert \Delta u \vert ^{2}\,dx + \frac{c^{2} \vert \beta \vert ^{2}}{4} \int _{\Omega }u^{4} \vert \nabla u \vert ^{2}\,dx \\ &\qquad{}+\frac{c_{6}}{2} \Vert \nabla u \Vert ^{2}. \end{aligned} $$

By Nirenberg’s inequality, we obtain

$$\begin{aligned}& \Vert u \Vert _{4}\leq c_{1}' \Vert \nabla \Delta u \Vert ^{\frac{1}{6}} \Vert u \Vert ^{\frac{5}{6}}+c_{2}' \Vert u \Vert ,\qquad \Vert \nabla u \Vert _{4}\leq c_{1}' \Vert \nabla \Delta u \Vert ^{\frac{1}{2}} \Vert u \Vert ^{\frac{1}{2}}+c_{2}' \Vert u \Vert , \\& \Vert u \Vert _{8}\leq c_{1}' \Vert \nabla \Delta u \Vert ^{\frac{1}{4}} \Vert u \Vert ^{\frac{3}{4}}+c_{2}' \Vert u \Vert ,\qquad \Vert \Delta u \Vert _{4}\leq c_{1}' \Vert \nabla \Delta u \Vert ^{\frac{5}{6}} \Vert u \Vert ^{\frac{1}{6}}+c_{2}' \Vert u \Vert . \end{aligned}$$

Thus, by Hölder’s inequality and the above inequalities, we deduce that

$$ \frac{c}{2} \int _{\Omega } \vert \nabla u \vert ^{4}\,dx+ \frac{c}{2} \int _{\Omega } \vert u \Delta u \vert ^{2}\,dx + \frac{c^{2} \vert \beta \vert ^{2}}{4} \int _{\Omega }u^{4} \vert \nabla u \vert ^{2}\,dx \leq \frac{\gamma }{2} \Vert \nabla \Delta u \Vert ^{2}+\frac{c_{7}}{2}. $$

Summing up, we obtain

$$ \frac{d}{dt} \Vert \nabla u \Vert ^{2}+\gamma \Vert \nabla \Delta u \Vert ^{2}\leq c_{6} \Vert \nabla u \Vert ^{2}+c_{7}. $$
(10)

On the other hand,

$$ \Vert \nabla u \Vert ^{2}=- \int _{\Omega }u\Delta u\,dx\leq \Vert u \Vert \Vert \Delta u \Vert \leq \sqrt{\frac{2c_{3}(c')^{2}}{2\gamma -c_{2}(c')^{2}}} \Vert \Delta u \Vert $$

and

$$ \Vert \Delta u \Vert ^{2}=- \int _{\Omega }\nabla u\cdot \nabla \Delta u\,dx\leq \Vert \nabla u \Vert \Vert \nabla \Delta u \Vert . $$

Adding the above two inequalities together gives

$$ c_{6} \Vert \nabla u \Vert ^{2}\leq c \Vert \nabla \Delta u \Vert ^{\frac{4}{3}}\leq \frac{\gamma }{2} \Vert \nabla \Delta u \Vert ^{2}+c_{8}. $$
(11)

It then follows from (10) and (11) that

$$ \frac{d}{dt} \Vert \nabla u \Vert ^{2}+\frac{\gamma }{2} \Vert \nabla u \Vert ^{2}\leq c_{7}+c_{8}. $$

Applying Gronwall’s inequality yields

$$ \Vert \nabla u \Vert ^{2}\leq e^{-\frac{\gamma }{2}t} \Vert \nabla u_{0} \Vert ^{2}+ \frac{2(c_{7}+c_{8})}{\gamma }\leq \frac{4(c_{7}+c_{8})}{\gamma } $$
(12)

for all \(t\geq T'=\max \{T^{*},\frac{2}{\gamma }\ln \frac{\gamma R^{2}}{2(c_{7}+c_{8})}\}\). Integrating (10) over \((t,t+1)\) with \(t\geq T'\) gives

$$\begin{aligned} \int _{t}^{t+1} \Vert \nabla \Delta u \Vert ^{2}\,d\tau \leq c_{9}. \end{aligned}$$

Using a mean value theorem for integrals, we obtain the existence of a time \(t_{0}\in (T',T'+1)\) such that

$$ \bigl\Vert \nabla \Delta u(t_{0}) \bigr\Vert ^{2}\leq c_{10} $$

holds uniformly. Since we consider problem (1)–(3) in the 2D case, based on Sobolev’s embedding theorem, we can get

$$ \Vert u \Vert _{p}= \biggl( \int _{\Omega }u^{p}\,dx \biggr)^{\frac{1}{p}}\leq c_{11}, \quad 1\leq p< \infty . $$

Set \(T_{1}=T'\), we complete the proof. □

Lemma 2.3

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{2}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)(3), we have

$$ \bigl\Vert Au(t) \bigr\Vert \leq M_{2}, \quad \forall t\geq T_{2}, $$

and

$$ \int _{t}^{t+1} \Vert u_{t} \Vert ^{2}\,d\tau \leq M_{2}, \quad t\geq T_{2}. $$

Here, \(M_{2}\) is a positive constant depending on γ and \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{2}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|^{2}_{H^{1}_{per}}\leq R^{2}\).

Proof

Multiplying equation (1) by \(\Delta ^{2}u\) and integrating the resulting relation over Ω, we obtain

$$ \begin{aligned} & \frac{1}{2}\frac{d}{dt} \Vert \Delta u \Vert ^{2}+\gamma \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2} \\ &\quad =\bigl(\Delta \varphi (u),\Delta ^{2}u\bigr)+\bigl(\beta \cdot \nabla \psi (u), \Delta ^{2}u\bigr) \\ &\quad =\bigl(\varphi '(u)\Delta u+\varphi {''}(u) \vert \nabla u \vert ^{2},\Delta ^{2}u\bigr)+ \beta \cdot \bigl(\psi (u)\nabla u,\Delta ^{2}u\bigr) \\ &\quad \leq \frac{\gamma }{2} \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2}+\frac{2}{\gamma } \bigl\Vert \varphi '(u) \Delta u \bigr\Vert ^{2}+\frac{2}{\gamma } \bigl\Vert \varphi {''}(u) \vert \nabla u \vert ^{2} \bigr\Vert ^{2} + \frac{ \vert \beta \vert ^{2}}{\gamma } \bigl\Vert \psi '(u) \nabla u \bigr\Vert ^{2}. \end{aligned} $$

Simple calculation shows that

$$ \begin{aligned} & \frac{d}{dt} \Vert \Delta u \Vert ^{2}+\gamma \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2} \\ &\quad \leq \frac{4}{\gamma } \int _{\Omega } \bigl\vert \varphi '(u)\Delta u \bigr\vert ^{2}\,dx+ \frac{4}{\gamma } \int _{\Omega } \bigl\vert \varphi {''}(u) \vert \nabla u \vert ^{2} \bigr\vert ^{2}\,dx+c \int _{\Omega }u^{2} \bigl\vert \varphi '(u) \bigr\vert \vert \nabla u \vert ^{2}\,dx+c \Vert \nabla u \Vert ^{2} \\ &\quad \leq c \biggl( \int _{\Omega }u^{4} \vert \Delta u \vert ^{2}\,dx+ \int _{\Omega }u^{2} \vert \nabla u \vert ^{4}\,dx+ \int _{\Omega }u^{6} \vert \nabla u \vert ^{2}\,dx \biggr)\\ &\qquad {}+c \Vert \Delta u \Vert ^{2}+c \Vert \nabla u \Vert _{4}^{4}+c \Vert \nabla u \Vert ^{2} \\ &\quad \leq c\bigl( \Vert u \Vert _{8}^{4}\|\Delta u \|_{4}^{2}+\|u\|_{4}^{2}\|\nabla u \|^{4}_{8}+ \|u\|_{12}^{6}\|\nabla u|_{4}^{2}\bigr)+c \Vert \Delta u \Vert ^{2}+c \Vert \nabla u \Vert _{4}^{4}+c \\ &\quad \leq c\bigl( \Vert \Delta u \Vert _{4}^{2}+ \Vert \nabla u \Vert ^{4}_{8}++ \Vert \nabla u \Vert _{4}^{2}+ \Vert \nabla u \Vert _{4}^{4} \bigr)+c \Vert \Delta u \Vert ^{2}+c . \end{aligned} $$

By Sobolev’s embedding theorem, we deduce that

$$\begin{aligned}& \Vert \Delta u \Vert _{4}^{2}\leq \bigl(c_{1}' \bigl\Vert \Delta ^{2}u \bigr\Vert ^{\frac{5}{8}} \Vert u \Vert ^{ \frac{3}{8}}+c_{2}' \Vert u \Vert \bigr)^{2}\leq \frac{\varepsilon }{c} \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2}+c_{ \varepsilon }, \\& \Vert \nabla u \Vert _{4}^{2}\leq \bigl(c_{1}' \bigl\Vert \Delta ^{2}u \bigr\Vert ^{\frac{3}{8}} \Vert u \Vert ^{ \frac{5}{8}}+c_{2}' \Vert u \Vert \bigr)^{2}\leq \frac{\varepsilon }{c} \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2}+c_{ \varepsilon }, \\& \Vert \nabla u \Vert _{4}^{4}\leq \bigl(c_{1}' \bigl\Vert \Delta ^{2}u \bigr\Vert ^{\frac{3}{8}} \Vert u \Vert ^{ \frac{5}{8}}+c_{2}' \Vert u \Vert \bigr)^{4}\leq \frac{\varepsilon }{c} \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2}+c_{ \varepsilon }, \end{aligned}$$

and

$$ \Vert \nabla u \Vert _{8}^{4}\leq \bigl(c_{1}' \bigl\Vert \Delta ^{2}u \bigr\Vert ^{\frac{7}{16}} \Vert u \Vert ^{ \frac{11}{16}}+c_{2}' \Vert u \Vert \bigr)^{2}\leq \frac{\varepsilon }{c} \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2}+c_{\varepsilon }. $$

Moreover,

$$ c \Vert \Delta u \Vert ^{2}=-c \int _{\Omega }\nabla u\cdot \nabla \Delta u\,dx=c \int _{\Omega }u\Delta ^{2}u\,dx\leq \Vert u \Vert \bigl\Vert \Delta ^{2}u \bigr\Vert \leq \varepsilon \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2}+c_{\varepsilon }. $$

Summing up and setting \(\varepsilon =\frac{\gamma }{10}\) gives

$$ \frac{d}{dt} \Vert \Delta u \Vert ^{2}+ \frac{\gamma }{2} \bigl\Vert \Delta ^{2}u \bigr\Vert ^{2} \leq c_{12}. $$
(13)

By a Calderón–Zygmund type estimate, the following inequality holds:

$$ \frac{d}{dt} \Vert \Delta u \Vert ^{2}+\frac{\gamma c'}{2} \bigl( \Vert \Delta u \Vert ^{2}+ \Vert \nabla \Delta u \Vert ^{2}\bigr)\leq c_{12}. $$

Then, using Gronwall’s inequality, we obtain

$$ \Vert \Delta u \Vert ^{2}\leq e^{-\frac{\gamma c'}{2}(t-t_{0}')} \bigl\Vert \Delta u\bigl(t'_{0}\bigr) \bigr\Vert ^{2}+\frac{2c_{12}}{\gamma c'}\leq \frac{4c_{12}}{\gamma c'} $$
(14)

for all \(t\geq T_{0}'=\max \{T_{0},t_{0}'+\frac{2}{\gamma c'}\ln \frac{\gamma c'R^{2}}{2c_{12}}\}\). Setting \(t\geq T'_{0}\), taking \(s\in (t,t+1)\), integrating (14) over \((s,t+1)\), we derive that

$$ \bigl\Vert \Delta u(t+1) \bigr\Vert ^{2}\leq c_{13}+ \bigl\Vert \Delta u(s) \bigr\Vert ^{2}. $$
(15)

Integrating (15) with respect to s in \((t,t+1)\), we can obtain

$$ \bigl\Vert \Delta u(t+1) \bigr\Vert ^{2}\leq c_{13}+ \int _{t}^{t+1} \bigl\Vert \Delta u(s) \bigr\Vert ^{2}\,dx \leq c_{14},\quad \forall t\geq T'_{0}. $$
(16)

By (14), (12), (7), and Sobolev’s embedding theorem, we conclude

$$ \Vert u \Vert _{\infty }\leq c_{15}, \qquad \Vert \nabla u \Vert _{p}\leq c_{16}, \quad 1\leq p< \infty . $$
(17)

Multiplying equation (1) by \(u_{t}\), integrating the resulting relation over Ω yields

$$ \begin{aligned} & \Vert u_{t} \Vert ^{2}+ \frac{\gamma }{2}\frac{d}{dt} \Vert \Delta u \Vert ^{2} \\ &\quad = \int _{\Omega }\Delta \varphi (u)u_{t}\,dx+\beta \cdot \int _{\Omega } \nabla \psi (u)u_{t}\,dx \\ &\quad = \int _{\Omega }\varphi '(u)\Delta uu_{t} \,dx+ \int _{\Omega }\varphi {''}(u) \vert \nabla u \vert ^{2}u_{t}\,dx+\beta \cdot \int _{\Omega }\psi '(u)\nabla uu_{t}\,dx \\ &\quad \leq \bigl\Vert \varphi '(u) \bigr\Vert _{\infty } \Vert \Delta u \Vert \Vert u_{t} \Vert + \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{4}^{2} \Vert u_{t} \Vert + \vert \beta \vert \bigl\Vert \psi '(u) \bigr\Vert _{ \infty } \Vert \nabla u \Vert \Vert u_{t} \Vert \\ &\quad \leq \frac{1}{2} \Vert u_{t} \Vert ^{2}+c \bigl( \bigl\Vert \varphi '(u) \bigr\Vert _{\infty }^{2} \Vert \Delta u \Vert ^{2}+ \bigl\Vert \varphi {''}(u) \bigr\Vert ^{2}_{\infty } \Vert \nabla u \Vert _{4}^{4}+ \vert \beta \vert ^{2} \bigl\Vert \psi '(u) \bigr\Vert ^{2}_{\infty } \Vert \nabla u \Vert ^{2}\bigr) \\ &\quad \leq \frac{1}{2} \Vert u_{t} \Vert ^{2}+ \frac{c_{17}}{2}, \end{aligned} $$

that is,

$$ \Vert u_{t} \Vert ^{2}+\frac{d}{dt} \Vert \Delta u \Vert ^{2}\leq c_{17}. $$
(18)

Integrating (18) over \((t+1,t+2)\), using (14), we derive that

$$ \int _{t+1}^{t+2} \Vert u_{t} \Vert ^{2}\,dx\leq c_{18},\quad \forall t\geq T_{0}^{\prime\prime}. $$

Using a mean value theorem for integrals, we obtain the existence of a time \(t_{1}\in (T_{0}^{\prime\prime}+1,T_{0}^{\prime\prime}+2)\) such that the following estimate holds uniformly:

$$ \bigl\Vert u_{t}(t_{1}) \bigr\Vert ^{2}\leq c_{19}. $$

Then the proof is complete. □

Lemma 2.4

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)(3), we have

$$ \bigl\Vert \nabla \Delta u(t) \bigr\Vert \leq M_{3}, \quad \forall t\geq T_{3}, $$

and

$$ \int _{t}^{t+1} \bigl\Vert A^{\frac{1}{2}}u_{t}(t) \bigr\Vert ^{2}\,dt\leq M_{3}, \quad \forall t \geq T_{3}. $$

Here, \(M_{3}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{3}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}}^{2}\leq R^{2}\).

Proof

Multiplying (1) by \(\Delta ^{3}u\) and integrating the resulting relation over Ω, we obtain

$$ \begin{aligned} & \frac{1}{2}\frac{d}{dt} \Vert \nabla \Delta u \Vert ^{2}+\gamma \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert ^{2} \\ &\quad = \int _{\Omega }\nabla \Delta \varphi (u)\nabla \Delta ^{2}u\,dx+\beta \cdot \int _{\Omega }\Delta \psi (u)\nabla \Delta ^{2}u\,dx \\ &\quad = \int _{\Omega }\varphi '(u)\nabla \Delta u\nabla \Delta ^{2}u\,dx+3 \int _{\Omega }\varphi {''}(u)\nabla u \Delta u\nabla \Delta ^{2}u\,dx\\ &\qquad {}+ \int _{\Omega }\varphi {'''}(u) \vert \nabla u \vert ^{2}\nabla u\nabla \Delta ^{2}u\,dx \\ &\qquad{}+\beta \cdot \int _{\Omega }\psi '(u)\Delta u\nabla \Delta ^{2}u\,dx+ \beta \cdot \int _{\Omega }\psi {''}(u) \vert \nabla u \vert ^{2}\nabla \Delta ^{2}u\,dx \\ &\quad \leq \frac{\gamma }{2} \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert ^{2}+c\bigl( \bigl\Vert \varphi '(u) \bigr\Vert _{ \infty }^{2} \Vert \nabla \Delta u \Vert ^{2}+3 \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty }^{2} \Vert \nabla u \Vert _{4}^{2} \Vert \Delta u \Vert _{4}^{2} \\ &\qquad{}+ \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty }^{2} \Vert \nabla u \Vert _{6}^{6}+ \vert \beta \vert ^{2} \bigl\Vert \psi '(u) \bigr\Vert _{\infty }^{2} \Vert \Delta u \Vert ^{2}+ \vert \beta \vert ^{2}\bigl\| \psi {''}(u) \bigr\| _{\infty }^{2}\|\nabla u\|_{4}^{4}\bigr). \end{aligned} $$
(19)

It follows form (17) that

$$\begin{aligned}& \bigl\Vert \varphi '(u) \bigr\Vert _{\infty }^{2} \Vert \nabla \Delta u \Vert ^{2}\leq \frac{c'}{2} \Vert \nabla \Delta u \Vert ^{2}, \\& 3 \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty }^{2} \Vert \nabla u \Vert _{4}^{2} \Vert \Delta u \Vert _{4}^{2} \leq \frac{c'}{2} \Vert \Delta u \Vert _{4}^{2}, \end{aligned}$$

and

$$ \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty }^{2} \Vert \nabla u \Vert _{6}^{6}+ \vert \beta \vert ^{2} \bigl\Vert \psi '(u) \bigr\Vert _{\infty }^{2} \Vert \|\Delta u \Vert ^{2}+ \vert \beta \vert ^{2}\bigl\| \psi {''}(u) \bigr\| _{\infty }^{2}\|\nabla u\|_{4}^{4}\leq \frac{c_{19}}{2}. $$

Summing up, we find that

$$ \frac{d}{dt} \Vert \nabla \Delta u \Vert ^{2}+ \gamma \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert ^{2} \leq c'\bigl( \Vert \nabla \Delta u \Vert ^{2}+ \Vert \Delta u \Vert _{4}^{2}+c_{19}\bigr). $$
(20)

Using Nirenberg’s inequality, we obtain

$$ c' \Vert \Delta u \Vert _{4}^{2}\leq c'\bigl(c_{1}' \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert ^{\frac{1}{6}} \Vert \Delta u \Vert ^{\frac{5}{6}}+c_{2}' \Vert \Delta u \Vert \bigr)^{2}\leq \frac{\gamma }{4} \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert ^{2}+c_{20}. $$

On the other hand,

$$ c' \Vert \nabla \Delta u \Vert ^{2}=c' \int _{\Omega }\nabla u\cdot \nabla \Delta ^{2}u\,dx\leq c' \Vert \nabla u \Vert \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert \leq \frac{\gamma }{4} \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert ^{2}+c_{21}. $$

Hence

$$ \frac{d}{dt} \Vert \nabla \Delta u \Vert ^{2}+ \frac{\gamma }{2} \bigl\Vert \nabla \Delta ^{2}u \bigr\Vert ^{2}\leq c_{20}+c_{21}+c'c_{19}. $$
(21)

A simple calculation shows that

$$ \frac{d}{dt} \Vert \nabla \Delta u \Vert ^{2}+c_{22} \Vert \nabla \Delta u \Vert ^{2} \leq c_{23}. $$
(22)

By Gronwall’s inequality, we immediately obtain

$$ \begin{aligned} \bigl\Vert \nabla \Delta u(t) \bigr\Vert ^{2}&\leq e^{-c_{22} (t-t_{0})} \bigl\Vert \nabla u(t_{0}) \bigr\Vert ^{2}+\frac{c_{23}}{c_{22}}\leq \frac{2c_{23}}{c_{22}} \end{aligned} $$
(23)

for all \(t\geq T_{1}^{*}=\max \{T_{1},t_{0}+\frac{1}{c_{22}}\ln \frac{c_{22}R^{2}}{2c_{23}}\}\). Combining (23), (14), (12), and (7) together gives

$$\begin{aligned} \Vert \nabla u \Vert _{\infty }\leq c_{24}, \qquad \Vert \Delta u \Vert _{q}\leq c_{25}, \quad 1 \leq q< \infty , \forall t\geq T_{1}^{*}. \end{aligned}$$
(24)

Multiplying equation (1) by \(Au_{t}\), integrating the resulting relation over Ω, we obtain

$$ \begin{aligned} & \Vert \nabla u_{t} \Vert ^{2}+\frac{\gamma }{2}\frac{d}{dt} \Vert \nabla \Delta u \Vert ^{2} \\ &\quad = \int _{\Omega }\nabla \Delta \varphi (u)\nabla u_{t}\,dx+ \beta \cdot \int _{\Omega }\Delta \varphi (u)\nabla u_{t}\,dx \\ &\quad = \int _{\Omega }\bigl[\varphi '(u)\nabla \Delta u+3 \varphi {''}(u)\nabla u \Delta u+\varphi {'''}(u) \vert \nabla u \vert ^{2}\nabla u\bigr]\nabla u_{t}\,dx \\ &\qquad{}+\beta \cdot \int _{\Omega }\bigl[\psi '(u)\Delta u+\psi {''}(u) \vert \nabla u \vert ^{2}\bigr] \nabla u_{t}\,dx \\ &\quad \leq \bigl\Vert \varphi '(u) \bigr\Vert _{\infty } \Vert \nabla \Delta u \Vert \Vert \nabla u_{t} \Vert +3 \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty } \Vert \Delta u \Vert \Vert \nabla u_{t} \Vert \\ &\qquad{}+ \bigl\Vert \varphi {'''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty }^{2} \Vert \nabla u \Vert \Vert \nabla u_{t} \Vert \\ &\qquad{}+ \vert \beta \vert \bigl\Vert \psi '(u) \bigr\Vert _{\infty } \Vert \Delta u \Vert \Vert \nabla u_{t} \Vert + \vert \beta \vert \bigl\Vert \psi {''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty } \Vert \nabla u \Vert \Vert \nabla u_{t} \Vert \\ &\quad \leq c \Vert \nabla u_{t} \Vert \leq \frac{1}{2} \Vert \nabla u_{t} \Vert ^{2}+ \frac{c_{26}}{2}. \end{aligned} $$

Summing up, using the result of (23) gives

$$ \Vert \nabla u_{t} \Vert ^{2}+\gamma \frac{d}{dt} \Vert \nabla \Delta u \Vert ^{2}\leq c_{26}. $$
(25)

Then

$$ \gamma \frac{d}{dt} \Vert \nabla \Delta u \Vert ^{2}\leq c_{26}. $$

Setting \(t\geq T_{1}^{*}\), taking \(s\in (t,t+1)\), integrating the above inequality over \((s,t+1)\), we obtain

$$ \bigl\Vert \nabla \Delta u(t+1) \bigr\Vert ^{2}\leq \frac{1}{\gamma }\bigl(c_{26}+ \bigl\Vert \nabla \Delta u(s) \bigr\Vert ^{2}\bigr). $$

Integrating the above inequality with respect to s in \((t,t+1)\), we have

$$\begin{aligned} \bigl\Vert \nabla \Delta u(t+1) \bigr\Vert ^{2}\leq \frac{1}{\gamma } \biggl(c_{26}+ \int _{t}^{t+1} \bigl\Vert \nabla \Delta u(s) \bigr\Vert ^{2}\,ds \biggr)\leq c_{27}, \quad \forall t\geq T_{1}^{*}. \end{aligned}$$
(26)

Integrating (25) over \((t+1,t+2)\), using (26) yields

$$ \int _{t+1}^{t+2} \bigl\Vert A^{\frac{1}{2}}u_{t} \bigr\Vert ^{2}\,d\tau \leq c_{28}, \quad \forall t \geq T_{1}^{*}. $$

Using a mean value theorem for integrals, we obtain the existence of a time \(t_{2}\in (T_{1}^{*}+1,T_{1}^{*}+2)\) such that the following estimate holds uniformly:

$$ \bigl\Vert A^{\frac{1}{2}}u_{t}(t_{2}) \bigr\Vert ^{2}\leq c_{29}. $$

Then we complete the proof. □

Lemma 2.5

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)(3), we have

$$ \Vert u_{t} \Vert \leq M_{4}, \quad \forall t\geq T_{4}. $$

Here, \(M_{4}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{4}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).

Proof

Setting \(v=u_{t}\), differentiating (1) with respect to the time t, we deduce that

$$ v_{t}+\gamma \Delta ^{2}v-\bigl[\Delta \varphi (u)\bigr]_{t}-\beta \cdot \bigl[ \nabla \psi (u) \bigr]_{t}=0. $$
(27)

Multiplying (27) by v, integrating the resulting relation over Ω yields

$$ \frac{1}{2}\frac{d}{dt} \Vert v \Vert ^{2}+\gamma \Vert \Delta v \Vert ^{2}- \int _{\Omega }\bigl[ \Delta \varphi (u)\bigr]_{t}v\,dx- \int _{\Omega }\beta \cdot \bigl[\nabla \psi (u) \bigr]_{t}v\,dx=0. $$
(28)

Using Sobolev’s embedding theorem, we get

$$\begin{aligned} &\int _{\Omega }\bigl[\Delta \varphi (u)\bigr]_{t}v\,dx+ \beta \cdot \int _{\Omega }\bigl[ \psi '(u)\nabla u \bigr]_{t}v\,dx \\ &\quad = \int _{\Omega }\varphi '(u)v\Delta v\,dx+ \int _{\Omega }\varphi {''}(u)v^{2} \Delta u\,dx+ \int _{\Omega }\varphi {'''}(u) \vert \nabla u \vert ^{2}v^{2}\,dx \\ &\qquad{}+2 \int _{\Omega }\varphi {''}(u)v\nabla u \nabla v\,dx +\beta \cdot \int _{ \Omega }\psi '(u)v\nabla v\,dx+\beta \cdot \int _{\Omega }\psi {''}(u)v^{2} \nabla u\,dx \\ &\quad \leq \bigl\Vert \varphi '(u) \bigr\Vert _{\infty } \Vert \Delta v \Vert \Vert v \Vert + \bigl\Vert \varphi {''}(u) \bigr\Vert _{ \infty } \Vert \Delta u \Vert _{6} \Vert v \Vert _{6}^{2}+ \bigl\Vert \varphi {'''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty }^{2} \Vert v \Vert ^{2} \\ &\qquad{}+2 \bigl\Vert \varphi {''}(u)\nabla u \bigr\Vert _{\infty } \Vert v \Vert \Vert \nabla v \Vert + \vert \beta \vert \bigl\Vert \psi '(u) \bigr\Vert _{\infty } \Vert \nabla v \Vert \Vert v \Vert + \vert \beta \vert \bigl\Vert \psi {''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty } \Vert v \Vert ^{2} \\ &\quad \leq c\bigl( \Vert \Delta v \Vert \Vert v \Vert + \Vert \nabla v \Vert ^{2}+ \Vert \nabla v \Vert \Vert v \Vert \bigr) \\ &\quad \leq \frac{\gamma }{2} \Vert \Delta v \Vert ^{2}+ \frac{c_{30}}{2} \Vert v \Vert ^{2}+ \frac{c_{31}}{2}. \end{aligned}$$

Hence,

$$ \frac{d}{dt} \Vert v \Vert ^{2}+\gamma \Vert \Delta v \Vert ^{2}\leq c_{30} \Vert v \Vert ^{2}+c_{31}. $$
(29)

A simple calculation shows that

$$ \Vert v \Vert ^{2}\leq \frac{1}{c'} \Vert \Delta v \Vert ^{2}. $$

It then follows from (29) and the above inequality that

$$ \frac{d}{dt} \Vert v \Vert ^{2}+ \bigl(c'\gamma -c_{30} \bigr) \Vert v \Vert ^{2}\leq c_{31}, $$

where γ is sufficiently large, it satisfies \(c'\gamma -c_{30}>0\). Using Gronwall’s inequality, we derive that

$$ \begin{aligned} \Vert v \Vert ^{2}&\leq e^{- (c'\gamma -c_{30} )(t-t_{1})} \bigl\Vert v(t_{1}) \bigr\Vert ^{2}+ \frac{c_{31}}{c'\gamma -c_{30}} \\ &\leq c_{19}e^{- (c'\gamma -c_{30} )(t-t_{1})} + \frac{c_{31}}{c'\gamma -c_{30}} \leq \frac{2c_{31}}{c'\gamma -c_{30}} \end{aligned} $$
(30)

for all \(t\geq t_{1}+\frac{1}{c'\gamma -c_{30}}\ln \frac{c_{19}(c'\gamma -c_{30})}{c_{31}}\). Then the proof is complete. □

Lemma 2.6

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)(3), we have

$$ \bigl\Vert A^{\frac{1}{2}}v_{t}(t) \bigr\Vert \leq M_{5}, \quad \forall t\geq T_{5}. $$

Here, \(M_{5}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{5}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).

Proof

Multiplying (27) by Av, integrating the resulting relation over Ω, we obtain

$$ \frac{1}{2}\frac{d}{dt} \Vert \nabla v \Vert ^{2}+ \gamma \Vert \nabla \Delta v \Vert ^{2}=- \int _{\Omega }\bigl[\Delta \varphi (u)\bigr]_{t}\Delta v \,dx-\beta \cdot \int _{ \Omega }\bigl[\nabla \psi (u)\bigr]_{t}\Delta v \,dx. $$
(31)

By Sobolev’s embedding theorem, we get

$$ \begin{aligned} &- \int _{\Omega }\bigl[\Delta \varphi (u)\bigr]_{t}\Delta v \,dx-\beta \cdot \int _{ \Omega }\bigl[\nabla \psi (u)\bigr]_{t}\Delta v \,dx \\ &\quad =- \int _{\Omega }\varphi '(u) \vert \Delta v \vert ^{2}\,dx- \int _{\Omega }\varphi {''}(u)v \Delta u \Delta v\,dx- \int _{\Omega }\varphi {'''}(u)v \vert \nabla u \vert ^{2} \Delta v\,dx \\ &\qquad{}+2 \int _{\Omega }\varphi {''}(u)\nabla u \nabla v\Delta v\,dx+\beta \cdot \int _{\Omega }\psi '(u)\nabla v\Delta v\,dx+\beta \cdot \int _{ \Omega }\psi {''}(u)v\nabla u\Delta v\,dx \\ &\quad \leq \bigl\Vert \varphi '(u) \bigr\Vert _{\infty } \Vert \Delta v \Vert ^{2}+ \bigl\Vert \varphi {''}(u) \bigr\Vert _{ \infty } \Vert \Delta u \Vert \Vert \Delta v \Vert \Vert v \Vert _{\infty }+ \bigl\Vert \varphi {'''}(u) \bigr\Vert _{ \infty } \Vert \nabla u \Vert _{\infty }^{2} \Vert \Delta v \Vert \Vert v \Vert \\ &\qquad{}+2 \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty } \Vert \nabla v \Vert \Vert \Delta v \Vert + \vert \beta \vert \bigl\Vert \psi '(u) \bigr\Vert _{\infty } \Vert \nabla v \Vert \Vert \Delta v \Vert \\ &\qquad{}+ \vert \beta \vert \bigl\Vert \psi {''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty } \Vert \Delta v \Vert \Vert v \Vert \\ &\quad \leq c\bigl( \Vert \Delta v \Vert ^{2}+ \Vert \Delta v \Vert \Vert v \Vert + \Vert \nabla v \Vert \Vert \Delta v \Vert \bigr) \leq \frac{\gamma }{2} \Vert \nabla \Delta v \Vert ^{2}+ \frac{c_{32}}{2} \Vert \nabla v \Vert ^{2}. \end{aligned} $$

Summing up gives

$$ \frac{d}{dt} \Vert \nabla v \Vert ^{2}+\gamma \Vert \nabla \Delta v \Vert ^{2}\leq c_{32} \Vert \nabla v \Vert ^{2}. $$

Using Nirenberg’s inequality, we obtain

$$ c_{32} \Vert \nabla v \Vert ^{2}\leq c_{32} \bigl(c_{1}' \Vert \nabla \Delta v \Vert ^{\frac{1}{3}} \Vert v \Vert ^{\frac{2}{3}}+c_{2}' \Vert v \Vert \bigr)^{2}\leq \frac{\gamma }{2} \Vert \nabla \Delta v \Vert ^{2}+c_{33}. $$

Adding the above two inequalities together gives

$$ \frac{d}{dt} \Vert \nabla v \Vert ^{2}+c_{32} \Vert \nabla v \Vert ^{2}\leq 2c_{33}. $$

By Gronwall’s inequality, we can obtain

$$ \begin{aligned} \Vert \nabla v \Vert ^{2}&\leq e^{-c_{32}(t-t_{2})} \bigl\Vert \nabla v(t_{2}) \bigr\Vert ^{2}+ \frac{2c_{33}}{c_{32}} \\ &\leq c_{29}e^{-c_{32}(t-t_{2})}+\frac{2c_{33}}{c_{32}}\leq \frac{4c_{33}}{c_{32}} \end{aligned} $$
(32)

for all \(t\geq t_{2}+\frac{1}{c_{32}}\ln \frac{c_{29}c_{32}}{2c_{33}}\). Then the proof is complete. □

Lemma 2.7

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)(3), we have

$$ \bigl\Vert A^{2}u(t) \bigr\Vert \leq M_{6}, \quad \forall t\geq T_{6}. $$

Here, \(M_{6}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{6}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).

Proof

For equation (1), by Lemmas 2.12.6, we deduce that

$$ \begin{aligned} \bigl\Vert \Delta ^{2} u \bigr\Vert & \leq \frac{1}{\gamma }\bigl( \Vert u_{t} \Vert + \bigl\Vert \Delta \varphi (u) \bigr\Vert + \vert \beta \vert \bigl\Vert \nabla \psi (u) \bigr\Vert \bigr) \\ &\leq c\bigl( \Vert u_{t} \Vert + \bigl\Vert \varphi '(u) \bigr\Vert _{\infty } \Vert \Delta u \Vert + \bigl\Vert \varphi {''}(u) \bigr\Vert _{\infty } \Vert \nabla u \Vert _{\infty } \Vert \nabla u \Vert + \vert \beta \vert \bigl\Vert \psi '(u) \bigr\Vert _{ \infty } \Vert \nabla u \Vert \bigr) \\ &\leq c_{34}, \quad \forall t\geq T. \end{aligned} $$

On the other hand, by Sobolev’s embedding theorem, it yields that

$$ \Vert \Delta u \Vert _{\infty }\leq c_{35}, $$

which completes the proof. □

Proof of Theorem 1.3

Suppose that \(M_{1}\) and \(M_{6}\) are the constants in Lemma 2.2 and Lemma 2.7, respectively. Denote

$$\begin{aligned}& B_{1}=\bigl\{ u\in \dot{H}^{1}_{per}: \bigl\Vert A^{\frac{1}{2}}u \bigr\Vert \leq M_{1}\bigr\} , \end{aligned}$$
(33)
$$\begin{aligned}& B_{2}=\bigl\{ u\in \dot{H}^{4}_{per}: \bigl\Vert A^{2}u \bigr\Vert \leq M_{6}\bigr\} . \end{aligned}$$
(34)

Using Lemmas 2.2 and 2.7, we easily obtain that \(B_{1}\) is a bounded \((\dot{H}^{1}_{per},\dot{H}_{per}^{1})\)-absorbing set for \(\{S(t)\}_{t\geq 0}\) and \(B_{2}\) is a bounded \((\dot{H}_{per}^{1},\dot{H}_{per}^{4})\)-absorbing set for \(\{S(t)\}_{t\geq 0}\). Note that the embedding \(\dot{H}^{4}_{per}\hookrightarrow \dot{H}^{1}_{per}\) is compacted. Applying Lemma 2.3, we obtain \(\{S(t)\}_{t\geq 0}\) is \((\dot{H}^{1}_{per},\dot{H}^{1}_{per})\)-asymptotically compact. Hence, \(\{S(t)\}_{t\geq 0}\) has an \((\dot{H}_{per}^{1}, \dot{H}_{per}^{1})\)-global attractor \(\mathcal{A}\). In the following, we show that \(\mathcal{A}\) is actually an \((\dot{H}_{per}^{1}, \dot{H}^{4}_{per})\)-global attractor for \(\{S(t)\}_{t\geq 0}\).

Lemma 3.1

Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy

$$ \varphi '(r)>0, \qquad \varphi ^{(i)}\leq c_{0}' \vert r \vert ^{k-i}+c_{1}', \qquad \psi '(r) \leq c_{0}r\sqrt{\varphi '(r)}+c_{1}, $$

where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for the solution \(u(x,t)\) of problem (1)(3), the dynamical system \(\{S(t)\}_{t\geq 0}\) is \((\dot{H}^{1}_{per}, \dot{H}^{4}_{per})\)-asymptotically compact.

Proof

For (1), we have

$$\begin{aligned} \gamma A^{2}u=-u_{t}+\Delta \varphi (u)+ \beta \cdot \nabla \psi (u). \end{aligned}$$
(35)

Assume that \(\{u_{0,n}\}_{n=1}^{\infty }\) is bounded in \(\dot{H}^{1}_{per}(\Omega )\) and \(t_{n}\rightarrow \infty \). In the following we prove that \(\{S(t_{n})u_{0,n}\}_{n=1}^{\infty }\) has a convergent subsequence in \(\dot{H}^{4}_{per}(\Omega )\). Denote

$$ u_{n}(t)=S(t)u_{0,n} \quad \text{and} \quad v_{n}(t_{n})=\frac{du_{n}}{dt}\bigg|_{t=t_{n}}. $$

Note that \(\{u_{0,n}\}_{n=1}^{\infty }\) is bounded in \(\dot{H}_{per}^{1}\). Then there exists \(R>0\) such that

$$ \bigl\Vert u_{0,n}+A^{\frac{1}{2}}u_{0,n} \bigr\Vert \leq R, \quad \forall n=1,2,\ldots . $$

By Lemmas 2.6 and 2.7, there exists \(T>0\) such that

$$\begin{aligned} \Vert v_{n} \Vert _{D(A^{\frac{1}{2}})}\leq M_{5}, \qquad \Vert u_{n} \Vert _{D(A^{2})} \leq M_{6}, \quad \forall t\geq T, n=1,2,\ldots . \end{aligned}$$
(36)

Since \(t_{n}\rightarrow \infty \), there exists \(N>0\) such that \(t_{n}\geq T\) for all \(n\geq N\). Therefore, by (36), we get

$$\begin{aligned} \bigl\Vert v_{n}(t_{n}) \bigr\Vert _{D(A^{\frac{1}{2}})}\leq M_{5}, \qquad \bigl\Vert u_{n}(t_{n}) \bigr\Vert _{D(A^{2})} \leq M_{6}, \quad \forall n\geq N. \end{aligned}$$
(37)

Note that the embedding \(D(A^{\frac{1}{2}})\hookrightarrow H\) and \(D(A^{2})\hookrightarrow D(A)\) are compacted. Hence, by (36), there exist \(v\in D(A^{\frac{1}{2}})\), \(\Delta u\in D(A)\), \(\nabla u\in \dot{H}^{3}_{per}\), and \(u\in \dot{H}_{per}^{4}\) such that, up to a subsequence,

$$\begin{aligned} \textstyle\begin{cases} v_{n}(t_{n})\rightarrow v \quad \text{strongly in } H, \\ \Delta u_{n}(t_{n})\rightarrow \Delta u \quad \text{strongly in } D(A^{ \frac{1}{2}}), \\ \nabla u_{n}(t_{n})\rightarrow \nabla u \quad \text{strongly in } D(A), \\ u_{n}(t_{n})\rightarrow u \quad \text{strongly in } \dot{H}_{per}^{3}. \end{cases}\displaystyle \end{aligned}$$
(38)

By (37) and Sobolev’s embedding theorem, we obtain

$$ \bigl\Vert u_{n}(t_{n}) \bigr\Vert _{W^{2,\infty }}\leq C, \quad \forall n\geq N. $$

It then follows from (36) and (38) that

$$\begin{aligned} \bigl\Vert u_{n}(t_{n})-u \bigr\Vert \rightarrow 0, \qquad \bigl\Vert v_{n}(t_{n})-v \bigr\Vert ^{2}\rightarrow 0, \qquad \bigl\Vert \Delta u_{n}(t_{n})- \Delta u \bigr\Vert ^{2}\rightarrow 0, \end{aligned}$$

and

$$ \begin{aligned} & \bigl\Vert \Delta \varphi \bigl(u_{n}(t_{n}) \bigr)-\Delta \varphi (u) \bigr\Vert \\ &\quad =\bigl\| \varphi '\bigl(u_{n}(t_{n})\bigr) \Delta u_{n}(t_{n})-\varphi '(u)\Delta u+ \varphi {''}\bigl(u_{n}(t_{n}) \bigr)\bigl|\nabla u_{n}(t_{n})) \bigr\vert ^{2}- \varphi {''}(u) \vert \nabla u \vert ^{2} \bigr\| \\ &\quad \leq c\bigl( \bigl\Vert \varphi '\bigl(u_{n}(t_{n}) \bigr)\bigl[\Delta u_{n}(t_{n})-\Delta u\bigr] \bigr\Vert + \bigl\Vert \Delta u\bigl[\varphi '\bigl(u_{n}(t_{n}) \bigr)\Delta u_{n}(t_{n})-\varphi '(u)\bigr] \bigr\Vert \\ &\qquad{}+\bigl\| \varphi {''}\bigl(u_{n}(t_{n}) \bigr)\bigl[\bigl|\nabla u_{n}(t_{n}) \bigr\vert ^{2}- \vert \nabla u \vert ^{2}\bigr] \bigr\Vert + \bigl\Vert \vert \nabla u \vert ^{2}\bigl[\varphi {''} \bigl(u_{n}(t_{n})\bigr)-\varphi {''}(u) \bigr] \bigr\Vert \bigr) \\ &\quad \leq c \bigl\Vert \varphi '\bigl(u_{n}(t_{n}) \bigr) \bigr\Vert _{\infty } \bigl\Vert \Delta u_{n}(t_{n})- \Delta u \bigr\Vert +c \Vert \Delta u \Vert _{\infty } \bigl\Vert \varphi '\bigl(u_{n}(t_{n})\bigr)\Delta u_{n}(t_{n})- \varphi '(u) \bigr\Vert \\ &\qquad{}+c \bigl\Vert \varphi {''} \bigl(u_{n}(t_{n})\bigr) \bigr\Vert _{\infty } \bigl\Vert \nabla u_{n}(t_{n})+ \nabla u \bigr\Vert _{\infty } \bigl\Vert \nabla u_{n}(t_{n})-\nabla u \bigr\Vert \\ &\qquad{}+c \Vert \nabla u \Vert ^{2}_{\infty } \bigl\Vert \varphi {''}\bigl(u_{n}(t_{n}) \bigr)-\varphi {''}(u) \bigr\Vert \\ &\quad \leq c \bigl\Vert \varphi '\bigl(u_{n}(t_{n}) \bigr) \bigr\Vert _{\infty } \bigl\Vert \Delta u_{n}(t_{n})- \Delta u \bigr\Vert \\ &\qquad{}+c \Vert \Delta u \Vert _{\infty } \bigl\Vert \varphi {''}\bigl(\theta _{1} u_{n}(t_{n})+(1- \theta _{1})u\bigr) \bigr\Vert _{\infty } \bigl\Vert u_{n}(t_{n})-u \bigr\Vert \\ &\qquad{}+c \bigl\Vert \varphi {''} \bigl(u_{n}(t_{n})\bigr) \bigr\Vert _{\infty } \bigl\Vert \nabla u_{n}(t_{n})+ \nabla u \bigr\Vert _{\infty } \bigl\Vert \nabla u_{n}(t_{n})-\nabla u \bigr\Vert \\ &\qquad{}+c \Vert \nabla u \Vert ^{2}_{\infty } \bigl\Vert \varphi {'''}\bigl(\theta _{2} u_{n}(t_{n})+(1- \theta _{2})u\bigr) \bigr\Vert _{\infty } \bigl\Vert u_{n}(t_{n})u \bigr\Vert \\ &\quad \leq c\bigl( \bigl\Vert \Delta u_{n}(t_{n})- \Delta u \bigr\Vert + \bigl\Vert \nabla u_{n}(t_{n})- \nabla u \bigr\Vert + \bigl\Vert u_{n}(t_{n})-u \bigr\Vert \bigr) \\ &\quad \rightarrow 0, \end{aligned} $$
(39)

where \(\theta _{1},\theta _{2}\in (0,1)\). Using the same method as above, we also have

$$ \bigl\Vert \nabla \psi \bigl(u_{n}(t_{n})\bigr)-\nabla \psi (u) \bigr\Vert \rightarrow 0. $$

Therefore

$$ \gamma A^{2}u_{n}(t_{n})\rightarrow -u_{t}+\Delta \varphi (u)+\beta \cdot \nabla \psi (u), \quad \text{strongly in }H, $$

that is, \(\{u_{n}(t_{n})\}_{n=1}^{\infty }\) converges to \(A^{-2}( -v+\Delta \varphi (u)+\beta \cdot \nabla \psi (u))\) in \(\dot{H}_{per}^{4}(\Omega )\). Then we complete the proof. □

Now we give the proof of the main result.

Proof of Theorem 1.3

Note that \(\{S(t)\}_{t\geq 0}\) has an \((\dot{H}^{1}_{per}, \dot{H}^{1}_{per})\)-global attractor \(\mathcal{A}\). By Lemma 2.7, \(B_{2}\) is a bounded \((\dot{H}^{1}_{per}, \dot{H}^{4}_{per})\)-absorbing set for \(\{S(t)\}_{t\geq 0}\). On the other hand, by Lemma 3.1, we can obtain \(\{S(t)\}_{t\geq 0}\) is \((\dot{H}_{per}^{1},\dot{H}_{per}^{4})\)-asymptotically compact. Then, by Proposition 1.2, \(\mathcal{A}\) is actually an \((\dot{H}^{1}_{per}, \dot{H}_{per}^{4})\)-global attractor for \(\{S(t)\}_{t\geq 0}\). The proof of Theorem 1.3 is complete. □

Availability of data and materials

All data generated or analysed during this study are included in this published article.

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Acknowledgements

The author would like to thank Dr. Ning Duan and Ms. Haichao Meng for their helpful suggestions.

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This paper was supported by the Fundamental Research Funds for the Central Universities (grant No. N2005031).

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The main idea of this paper was proposed by XZ. XZ prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

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Zhao, X. On global dynamics of 2D convective Cahn–Hilliard equation. Bound Value Probl 2020, 175 (2020). https://doi.org/10.1186/s13661-020-01477-3

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Keywords

  • Global attractor
  • Convective Cahn–Hilliard equation
  • Absorbing set
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