© Wuhan University 2023

0 Introduction

In this paper, we study the following three-dimensional Brinkman-Forchheimer equation:

$\{\begin{array}{c}{u}_t-\gamma \mathrm{\Delta }u+au+b\left|u\right|u+c{\left|u\right|}^{\beta }u+\nabla p=f(x),(x,t)\in \mathrm{\Omega }\times {\mathbb{R}}^{+}\\ \mathrm{d}\mathrm{i}\mathrm{v}u=\mathrm{0},(x,t)\in \mathrm{\Omega }\times {\mathbb{R}}^{+}\\ {u(x,t)|}_{\partial \mathrm{\Omega }}=\mathrm{0},(x,t)\in \partial \mathrm{\Omega }\times {\mathbb{R}}^{+}\\ u(x,\mathrm{0})=u_{\mathrm{0}}(x),x\in \mathrm{\Omega }\end{array}$(1)

where $a>\mathrm{0}$ is the Darcy coefficient, $b>\mathrm{0},c>\mathrm{0}$ are the Forchheimer coefficients, $\beta >\mathrm{0}$ is a constant, and $\mathrm{\Omega }\subseteq {\mathbb{R}}^{\mathrm{3}}$ is an open and bounded set, which is sufficiently regular. Here $u=(u_{\mathrm{1}}(x,t),u_{\mathrm{2}}(x,t),u_{\mathrm{3}}(x,t))$ is the fluid velocity vector, $p(x,t)$ denotes the pressure field, $f(x)$ is the external force, $\gamma$ is the Brinkman coefficient and $u_{\mathrm{0}}=u_{\mathrm{0}}(x)$ is the initial velocity.

As a mathematical model, Brinkman-Forchheimer equation describes the motion of a fluid flowing in saturated porous media ^[1-3], which has received much attention on several issues over the last decades. From a mathematical point of view, the research on the three-dimensional Brinkman-Forchheimer equation is mainly divided into two categories. One is the structural stability of the equation with respect to the coefficients $\gamma ,b$ and $c$^[4-9], and the other is the long-term behavior of the solution of the equation ^[10-18].

If a system has a global attractor, then the attractor will contain all possible limit states of the solutions of the system. Therefore, studying the dynamic system restricted to the global attractor will be able to reveal a lot of information about the original system. So proving the existence of global attractors is a basic and important problem in infinite-dimensional dynamical systems. For the 3D Brinkman-Forchheimer equation, Ugurlu ^[10], Ouyang and Yang ^[11] showed the existence of global attractor in ${\left(H_{\mathrm{0}}^{\mathrm{1}}(\mathrm{\Omega })\right)}^{\mathrm{3}}$ when $\beta =\mathrm{2}$ and $\mathrm{1}<\beta <\mathrm{4}/\mathrm{3}$ by condition-(C) method, respectively. Wang and Lin^[12] showed the existence of global attractor in $(H^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}$ when $\beta =\mathrm{2}$. In Ref.[13], the existence of $D$-pullback attractors for three-dimensional non-autonomous Brinkman-Forchheimer equation is deduced by establishing the $D$-pullback asymptotical compactness of $\theta$-cocycle. In Ref.[14], Song et al discussed the $L^{\mathrm{2}}$ decay of the weak solution of the Brinkman-Forchheimer equation in three-dimensional full space. In Ref.[15], Song and Wu investigated the uniform boundedness of uniform attractor $A^{\epsilon }$of equation (1) with singularly oscillating external force. They established the convergence of the attractor $A^{\epsilon }$ to the attractor $A^{\mathrm{0}}$ of the averaged equation as $\epsilon \to {\mathrm{0}}^{+}$. In Ref.[16], the pullback dynamics and asymptotic stability for a 3D Brinkman-Forchheimer equation with finite delay was concerned. In Ref.[17], by some estimates and the variable index to deal with the delay term, Yang et al got the sufficient conditions for asymptotic stability of trajectories inside the pullback attractors for a fluid flow model in porous medium by generalized Grashof numbers. In Ref.[18], Qiao et al proved the existence of a global attractor for the strong solution of the Brinkman-Forchheimer equation in a three-dimensional bounded domain.

In the geometric structure of the global attractor, the dimension is a very important property. This is because if the fractal dimension of the global attractor is finite, the original infinite-dimensional dynamical system can be reduced to a finite-dimensional ordinary differential equation system, so that the relatively complete theory of the finite-dimensional dynamical system can be used to study infinite dimensional dynamical system. As far as we know, there is no results on dimension estimate of global attractor of 3D Brinkman-Forchheimer. In this paper, inspired by Refs.[19-21], based on Ref.[18], we will discuss the Hausdorff dimension and fractal dimension of global attractors for strong solutions of the equation.

The structure of this paper is arranged as follows: In Section 1, we give some function space symbols and some inequalities that will be used later. In Section 2, we discuss the Hausdorff dimension and fractal dimension of global attractors for strong solution of the equation.

1 Preliminaries

In this section, we introduce some notations and preliminaries, which will be used throughout this paper.

First, let us introduce the following function spaces:

$E=\{u\in (C_{\mathrm{0}}^{\mathrm{\infty }}{(\mathrm{\Omega }))}^{\mathrm{3}}:\mathrm{d}\mathrm{i}\mathrm{v}u=\mathrm{0}\},H=\mathrm{c}{\mathrm{l}}_{(L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}}E,V=\mathrm{c}{\mathrm{l}}_{(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}}E,$

where $\mathrm{c}{\mathrm{l}}_X$ denotes the closure in space $X$. $H$ is the closure of the set $E$ in $(L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}$ topology, and $V$ is the closure of the set $E$ in $(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}$ topology. $H^{\text{'}}$ and $V^{\text{'}}$ are the dual spaces of $H$ and $V$, respectively. $H$ and $V$ are equipped with the following inner products:

$(u,v)={\int }_{\Omega }u\cdot vdx,\forall u,v\in H,((u,v))={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\nabla u_i\cdot \nabla v_i\mathrm{d}x,\forall u,v\in V,$

and norms $\Vert \cdot \Vert ={(\cdot ,\cdot )}^{\frac{\mathrm{1}}{\mathrm{2}}},{\Vert \cdot \Vert }_V=({(\cdot ,\cdot ))}^{\frac{\mathrm{1}}{\mathrm{2}}}.$ Let ${\mathit{L}}^p(\mathrm{\Omega })=(L^p{(\mathrm{\Omega }))}^{\mathrm{3}},{\mathit{H}}^{\mathrm{2}}(\mathrm{\Omega })=(H^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}.$ Throughout this paper, we use ${\Vert \cdot \Vert }_p$ to denote the norm in ${\mathit{L}}^p(\mathrm{\Omega })$. $C$ or $C_i$ will stand for some generic positive constants, depending on $\mathrm{\Omega }$ and some constants, but independent of time $t$.

We call $u\in L^{\mathrm{\infty }}(\mathrm{0},T;H)\bigcap L^{\mathrm{2}}(\mathrm{0},T;V)\bigcap L^{\beta +\mathrm{2}}(\mathrm{0},T;{\mathit{L}}^{\beta +\mathrm{2}}(\mathrm{\Omega }))$ is a weak solution of problem (1) on $[\mathrm{0},T]$, if

$\{\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}(u,v)+\gamma ((u,v))+a(u,v)+(b\left|u\right|u,v)+(c{\left|u\right|}^{\beta }u,v)=(f,v),\forall v\in V,t>\mathrm{0}\\ u(\mathrm{0})=u_{\mathrm{0}}\end{array}$(2)

The weak form (2) is equivalent to the following functional equation:

$\{\begin{array}{c}\frac{\mathrm{d}u}{\mathrm{d}t}+\gamma Au+au+B(u)=f,\forall t>\mathrm{0}\\ u(\mathrm{0})=u_{\mathrm{0}}\end{array}$(3)

Here $A=-P_H\mathrm{\Delta }$ is the Stokes operator subject to the no-slip homogeneous Dirichlet boundary condition with the domain $(H^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}\bigcap V$ defined as $〈Au,v〉=((u,v))$. $P_H$ is the orthogonal projection from ${\mathit{L}}^{\mathrm{2}}(\mathrm{\Omega })$ onto $H$. $F(u)=b\left|u\right|u+c{\left|u\right|}^{\beta }u,$$B(u)=P_{H}F(u),f\in H.$ Obviously, the operator $A$ is a non-negative self-adjoint operator in $H$ with $V=D(A^{\mathrm{1}/\mathrm{2}})$ and $〈Au,u〉={\Vert u\Vert }_V,$ for all $u\in V$.

For a bounded domain $\mathrm{\Omega }$, the operator $A$ is invertible, and its inverse operator $A^{-\mathrm{1}}$ is bounded, self-adjoint and compact in $H$. Thus, the spectrum of $A$ consists of an infinite sequence $\mathrm{0}<{\lambda }_{\mathrm{1}}\le {\lambda }_{\mathrm{2}}\le \cdots \le {\lambda }_k\le \cdots ,$ with ${\lambda }_k\to \mathrm{\infty }$ as its eigenvalue $k\to \mathrm{\infty }$ (Theorem 2.2, Corollary 2.2, Ref.[22]). For all $k\ge \mathrm{1}$ and $n\in \mathbb{N},$ we have ${\lambda }_k\ge \tilde{C}{k}^{\mathrm{2}/n},$ where $\tilde{C}=\frac{n}{\mathrm{2}+n}{\left(\frac{{(\mathrm{2}\mathrm{\pi })}^n}{{\omega }_n(n-\mathrm{1})\left|\mathrm{\Omega }\right|}\right)}^{\mathrm{2}/n},{\omega }_n={\mathrm{\pi }}^{n/\mathrm{2}}\mathrm{\Gamma }(\mathrm{1}+\frac{n}{\mathrm{2}}),$ and $\left|\mathrm{\Omega }\right|$ is the n-dimensional Lebesgue measure of $\mathrm{\Omega }$. For $n=\mathrm{3},$ we find

$\tilde{C}=\frac{{\mathrm{3}}^{\mathrm{1}/\mathrm{3}}{\mathrm{2}}^{\mathrm{8}/\mathrm{3}}{\mathrm{\pi }}^{\mathrm{2}/\mathrm{3}}}{\mathrm{5}{\left|\mathrm{\Omega }\right|}^{\mathrm{2}/\mathrm{3}}}$(4)

Next we formulate some well-known inequalities and a Gronwall type lemma that we will use in what follows.

Poincaré's inequality ^[23]:

${\Vert u\Vert }^{\mathrm{2}}\le \frac{\mathrm{1}}{{\lambda }_{\mathrm{1}}}{\Vert \nabla u\Vert }^{\mathrm{2}},\forall u\in V$(5)

where ${\lambda }_{\mathrm{1}}$ is the first eigenvalue of operator $A=-P_H\mathrm{\Delta }$ under the homogeneous Dirichlet boundary condition.

Agmon's inequality ^[23]:

${\Vert u\Vert }_{\mathrm{\infty }}\le C_{\mathrm{1}}{\Vert \nabla u\Vert }^{\mathrm{1}/\mathrm{2}}{\Vert \mathrm{\Delta }u\Vert }^{\mathrm{1}/\mathrm{2}},\forall u\in D(A)$(6)

$C_q$-inequality ^[24]:

$\left|x^q-y^q\right|\le C_q({\left|x\right|}^{q-\mathrm{1}}+{\left|y\right|}^{q-\mathrm{1}})\left|x-y\right|,fortheinteger[q\ge \mathrm{2}]$(7)

Series inequlity^[25]:

${\displaystyle \sum _{k=m}^n}{k}^p<\frac{\mathrm{1}}{p+\mathrm{1}}(n^{p+\mathrm{1}}-{(m-\mathrm{1})}^{p+\mathrm{1}}),for[-\mathrm{1}<p<\mathrm{0},n>m]$(8)

Lemma 1   (Gagliardo-Nirenberg's inequality) ^[24] Assuming that $\mathrm{\Omega }={\mathbb{R}}^n$ or $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded domain, which has a sufficiently smooth boundary $\partial \mathrm{\Omega }$, $u\in L^q(\mathrm{\Omega }),D^{m}u\in L^r(\mathrm{\Omega }),$$\mathrm{1}\le q,r\le \mathrm{\infty }$. Then there is a constant $C>\mathrm{0}$ such that ${\Vert D^{j}u\Vert }_p\le C{\Vert D^{m}u\Vert }_r^a{\Vert u\Vert }_q^{\mathrm{1}-a},$where $\frac{\mathrm{1}}{p}=\frac{j}{n}+a(\frac{\mathrm{1}}{r}-\frac{m}{n})+(\mathrm{1}-a)\frac{\mathrm{1}}{q},\mathrm{1}\le p\le \mathrm{\infty },\mathrm{0}\le j\le m,\frac{j}{m}\le a\le \mathrm{1},$$C$ depends on n,m,j,a,q,r.

Lemma 2   ( Gronwall's inequality) ^[23] Let $u(t),k(t)$ be non-negative integrable functions on $[\mathrm{0},T]$. If there is $K>\mathrm{0}$, such that $u(t)\le K+{\int }_{\mathrm{0}}^{t}k(s)u(s)ds,\forall t\in [\mathrm{0},T],$then $u(t)\le K\mathrm{e}\mathrm{x}\mathrm{p}({\int }_{\mathrm{0}}^{t}k(s)\mathrm{d}s),\forall t\in [\mathrm{0},T].$

Now we recall the existence and uniqueness theorem of the strong solutions of equation (1).

Theorem 1^[18]Suppose $\beta >\mathrm{0},u_{\mathrm{0}}\in V\bigcap {\mathit{L}}^{\beta +\mathrm{2}}(\mathrm{\Omega })$ and $f\in H$. Then there exists a strong solution of equation (1) satisfying

$u\in L^{\mathrm{\infty }}(\mathrm{0},T;V)\bigcap L^{\mathrm{\infty }}(\mathrm{0},T;{\mathit{L}}^{\beta +\mathrm{2}}(\mathrm{\Omega }))\bigcap L^{\mathrm{2}}(\mathrm{0},T;(H^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}),\nabla u{\left|u\right|}^{\beta /\mathrm{2}}\in L^{\mathrm{2}}(\mathrm{0},T;H),u_t\in L^{\mathrm{2}}(\mathrm{0},T;H).$

Moreover when $\mathrm{5}/\mathrm{2}\le \beta \le \mathrm{4},$ the strong solution is unique.

Now we will review the uniform estimates of strong solution to the problem (1) when $t\to \mathrm{\infty }$.

Lemma 3^[18] Suppose $\mathrm{5}/\mathrm{2}\le \beta \le \mathrm{4},u_{\mathrm{0}}\in V,f\in H.$ Then there exists a time $t_{\mathrm{0}}$, constants ${\rho }_{\mathrm{1}},I_{\mathrm{1}},$ such that when $t>t_{\mathrm{0}},$ we have $\Vert u(t)\Vert \le {\rho }_{\mathrm{1}},$${{\int }_t^{t+\mathrm{1}}\Vert u(s)\Vert }_V^{\mathrm{2}}\mathrm{d}s+{{\int }_t^{t+\mathrm{1}}\Vert u(s)\Vert }^{\mathrm{2}}\mathrm{d}s+{\int }_t^{t+\mathrm{1}}{\Vert u(s)\Vert }_{\mathrm{3}}^{\mathrm{3}}\mathrm{d}s+{{\int }_t^{t+\mathrm{1}}\Vert u(s)\Vert }_{\beta +\mathrm{2}}^{\beta +\mathrm{2}}\mathrm{d}s\le I_{\mathrm{1}},$for $t>t_{\mathrm{0}}$.

Lemma 4^[18]Suppose $\mathrm{5}/\mathrm{2}\le \beta \le \mathrm{4},u_{\mathrm{0}}\in V,f\in H.$ Then there exists a time $t_{\mathrm{1}}$,a constant ${\rho }_{\mathrm{2}},$ such that ${\Vert u(t)\Vert }_V+\Vert u(t)\Vert +{\Vert u(t)\Vert }_{\mathrm{3}}+{\Vert u(t)\Vert }_{\beta +\mathrm{2}}\le {\rho }_{\mathrm{2}},\forall t>t_{\mathrm{1}}.$

Lemma 5^[18] Suppose $\mathrm{5}/\mathrm{2}\le \beta \le \mathrm{4},u_{\mathrm{0}}\in V,f\in H.$ Then there exists a time $t_{\mathrm{2}}$,a constant ${\rho }_{\mathrm{3}},$ such that $\Vert u_t(s)\Vert \le {\rho }_{\mathrm{3}},\forall s\ge t_{\mathrm{2}}.$

Lemma 6^[18]Suppose $\mathrm{5}/\mathrm{2}\le \beta \le \mathrm{4},u_{\mathrm{0}}\in V,f\in H.$ Then there exists a constant ${\rho }_{\mathrm{4}},$ such that $\Vert Au(t)\Vert \le {\rho }_{\mathrm{4}},\forall t\ge t_{\mathrm{2}}.$

Finally, we give the result of existence of global attractor in $V$ for the 3D Brinkman-Forchheimer equation.

Theorem 2^[18]Suppose $\mathrm{5}/\mathrm{2}\le \beta \le \mathrm{4},u_{\mathrm{0}}\in V,f\in H.$ Then the problem (1) has a global attractor $A_V$ in $V$, which is invariant and compact in $V$ and attracts every bounded subset of $V$ with the norm in $V$.

2 Estimates of Dimensions of the Global Attractor

In this section, we will establish the differentiability of the semigroup with respect to the initial data. We show that the global attractor of the 3D Brinkman-Forchheimer system has finite Hausdorff and fractal dimensions. We will use the similar techniques as in Refs.[19-21], etc to obtain the desired results.

Let $u(\cdot )$ be the unique strong solution of the autonomous system (1) belonging to the global attractor $A_V$.

Let us take inner product with $-\mathrm{\Delta }u$ in $H$ to the first equation in (1) to obtain

$\frac{\mathrm{d}}{\mathrm{d}t}{\Vert u\Vert }_V^{\mathrm{2}}+\mathrm{2}\gamma \Vert \mathrm{\Delta }u\Vert {}_{}{}^{\mathrm{2}}+\mathrm{2}a{\Vert u\Vert }_V^{\mathrm{2}}+\mathrm{2}b{\int }_{\mathrm{\Omega }}\left|u\right|{\left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x+\frac{\mathrm{8}b}{\mathrm{9}}{{\int }_{\mathrm{\Omega }}\left|\nabla {\left|u\right|}^{\mathrm{3}/\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{d}x+\mathrm{2}c{{\int }_{\mathrm{\Omega }}\left|u\right|}^{\beta }{\left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x+\frac{\mathrm{8}c\beta }{{(\beta +\mathrm{2})}^{\mathrm{2}}}{{\int }_{\mathrm{\Omega }}\left|\nabla {\left|u\right|}^{\frac{\beta +\mathrm{2}}{\mathrm{2}}}\right|}^{\mathrm{2}}\mathrm{d}x$

$=-\mathrm{2}(f,\mathrm{\Delta }u)\le \mathrm{2}\Vert f\Vert \Vert \mathrm{\Delta }u\Vert \le \gamma {\Vert \mathrm{\Delta }u\Vert }^{\mathrm{2}}+\frac{\mathrm{1}}{\gamma }{\Vert f\Vert }^{\mathrm{2}}$(9)

So we have

$\frac{\mathrm{d}}{\mathrm{d}t}{\Vert u\Vert }_V^{\mathrm{2}}+\mathrm{2}a{\Vert u\Vert }_V^{\mathrm{2}}\le \frac{\mathrm{1}}{\gamma }{\Vert f\Vert }^{\mathrm{2}}$(10)

Applying Gronwall's inequality, we find

${\Vert u\Vert }_V^{\mathrm{2}}\le {\Vert u_{\mathrm{0}}\Vert }_V^{\mathrm{2}}{\mathrm{e}}^{-\mathrm{2}at}+\frac{{\Vert f\Vert }^{\mathrm{2}}}{\mathrm{2}a\gamma },\forall t>\mathrm{0}$(11)

So there is a time $t^{\text{'}}$ which we can take as $t^{\text{'}}=\mathrm{m}\mathrm{a}\mathrm{x}\{-\frac{\mathrm{1}}{\mathrm{2}a}\mathrm{l}\mathrm{n}\frac{{\Vert f\Vert }^{\mathrm{2}}}{\mathrm{2}a\gamma {\Vert u_{\mathrm{0}}\Vert }_V^{\mathrm{2}}},\mathrm{0}\}$, such that for all $t\ge t^{\text{'}}$, we have

${\Vert u\Vert }_V^{\mathrm{2}}\le \frac{{\Vert f\Vert }^{\mathrm{2}}}{a\gamma }=M_{\mathrm{1}}^{\mathrm{2}},M_{\mathrm{1}}=\frac{\Vert f\Vert }{\sqrt[]{a\gamma }}$(12)

Integrating the inequality (10) from 0 to $T$, we obtain

${\Vert u(T)\Vert }_V^{\mathrm{2}}+\mathrm{2}a{{\int }_{\mathrm{0}}^T\Vert u(s)\Vert }_V^{\mathrm{2}}\mathrm{d}s\le {\Vert u_{\mathrm{0}}\Vert }_V^{\mathrm{2}}+\frac{T}{\gamma }{\Vert f\Vert }^{\mathrm{2}}$(13)

so we get

$\underset{T\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}\frac{\mathrm{1}}{T}{{\int }_{\mathrm{0}}^T\Vert u(t)\Vert }_V^{\mathrm{2}}\mathrm{d}t\le K_{\mathrm{1}}=\frac{{\Vert f\Vert }^{\mathrm{2}}}{\mathrm{2}a\gamma }$(14)

Theorem 3   Let $u_{\mathrm{0}}$ and $v_{\mathrm{0}}$ be two members of $A_V$. Then there exists a constant $K=K({\Vert u_{\mathrm{0}}\Vert }_V,{\Vert v_{\mathrm{0}}\Vert }_V)$ such that

${\Vert S(t)u_{\mathrm{0}}-S(t)v_{\mathrm{0}}-\mathrm{\Lambda }(t)(u_{\mathrm{0}}-v_{\mathrm{0}})\Vert }_V\le K{\Vert u_{\mathrm{0}}-v_{\mathrm{0}}\Vert }_V$(15)

where the linear operator $\mathrm{\Lambda }(t):V\to V,$ for $t>\mathrm{0}$ is the solution operator of the problem:

$\{\begin{array}{c}\frac{\mathrm{d}\xi }{\mathrm{d}t}+\gamma A\xi +a\xi +\mathrm{2}b{P}_H(\left|u(t)\right|\xi (t))+c(\beta +\mathrm{1})P_H({\left|u(t)\right|}^{\beta }\xi (t))=\mathrm{0},t\in (\mathrm{0},T)\\ \xi (\mathrm{0})={\xi }_{\mathrm{0}}\in V\end{array}$(16)

${\xi }_{\mathrm{0}}=u_{\mathrm{0}}-v_{\mathrm{0}}$ and $u(t)=S(t)u_{\mathrm{0}}$, $v(t)=S(t)v_{\mathrm{0}}$. In other words, for every $t>\mathrm{0}$, the solution $S(t)u_{\mathrm{0}}$ as a map $S(t):V\to V$ is Fréchet differentiable for the initial data, and its Fréchet derivative $D_{u_{\mathrm{0}}}(S(t)u_{\mathrm{0}})w_{\mathrm{0}}=\mathrm{\Lambda }(t)w_{\mathrm{0}}$.

Proof   Let $S(t)u_{\mathrm{0}}=u(t),S(t)v_{\mathrm{0}}=v(t),$ for any $t\ge \mathrm{0}.$ Then we have

$\{\begin{array}{c}\frac{\mathrm{d}u}{\mathrm{d}t}+\gamma Au+au+P_H(b\left|u\right|u+c{\left|u\right|}^{\beta }u)=f\\ u(\mathrm{0})=u_{\mathrm{0}}\end{array}$(17)

and

$\{\begin{array}{c}\frac{\mathrm{d}v}{\mathrm{d}t}+\gamma Av+av+P_H(b\left|v\right|v+c{\left|v\right|}^{\beta }v)=f\\ v(\mathrm{0})=v_{\mathrm{0}}\end{array}$(18)

Let $w(t)=u(t)-v(t)$. Combining (17) with (18), we obtain

$\{\begin{array}{c}\frac{\mathrm{d}w}{\mathrm{d}t}+\gamma Aw+aw+P_H(b\left|u\right|u-b\left|v\right|v+c{\left|u\right|}^{\beta }u-c{\left|v\right|}^{\beta }v)=\mathrm{0}\\ w(\mathrm{0})=u_{\mathrm{0}}-v_{\mathrm{0}}\end{array}$(19)

Let us define $\eta (t)=u(t)-v(t)-\xi (t)=S(t)(u_{\mathrm{0}}-v_{\mathrm{0}})-\xi (t)$. Then $\eta (t)$ satisfies:

$\{\begin{array}{c}\begin{array}{l}\frac{\mathrm{d}\eta (t)}{\mathrm{d}t}+\gamma A\eta (t)+a\eta (t)+P_H(b\left|u(t)\right|u(t)-b\left|v(t)\right|v(t)+c{\left|u(t)\right|}^{\beta }u(t)-c{\left|v(t)\right|}^{\beta }v(t))\\ -\mathrm{2}b{P}_H(\left|u(t)\right|\xi (t))-c(\beta +\mathrm{1})P_H({\left|u(t)\right|}^{\beta }\xi (t))=\mathrm{0}\end{array}\\ \eta (\mathrm{0})=\mathrm{0}\end{array}$(20)

Let us take inner product with $A\eta (t)$ in $H$ to the first equation in (20) to obtain

$\begin{array}{l}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\Vert \nabla \eta (t)\Vert }^{\mathrm{2}}+\gamma {\Vert A\eta (t)\Vert }^{\mathrm{2}}+a{\Vert \nabla \eta (t)\Vert }^{\mathrm{2}}=\mathrm{2}b(P_H(\left|u(t)\right|w(t)),A\eta (t))-\mathrm{2}b(P_H(\left|u(t)\right|\eta (t)),A\eta (t))\\ +c(\beta +\mathrm{1})(P_H({\left|u(t)\right|}^{\beta }w(t)),A\eta (t))-c(\beta +\mathrm{1})(P_H({\left|u(t)\right|}^{\beta }\eta (t)),A\eta (t))\\ -(P_H(b\left|u(t)\right|u(t)-b\left|v(t)\right|v(t)+c{\left|u(t)\right|}^{\beta }u(t)-c{\left|v(t)\right|}^{\beta }v(t)),A\eta (t))\end{array}$(21)

Let us consider the first term on the right-hand side of (21). We have

$\begin{array}{l}\mathrm{2}b(P_H(\left|u(t)\right|w(t)),A\eta (t))\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{b}^{\mathrm{2}}}{\gamma }{{\int }_{\mathrm{\Omega }}\left|u(t)\right|}^{\mathrm{2}}{\left|w(t)\right|}^{\mathrm{2}}\mathrm{d}x\\ \le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{b}^{\mathrm{2}}}{\gamma }\Vert u(t)\Vert {}_{\mathrm{\infty }}{}^{\mathrm{2}}\Vert w(t)\Vert {\Vert w(t)\Vert }^{\mathrm{2}}\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{b}^{\mathrm{2}}{C}_{\mathrm{1}}^{\mathrm{2}}}{\gamma }\Vert \nabla u(t)\Vert \Vert \mathrm{\Delta }u(t)\Vert {\Vert w(t)\Vert }^{\mathrm{2}}\\ \le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\Vert w(t)\Vert }^{\mathrm{2}}\end{array}$(22)

In (22), Agmon's inequality is used. And in the last inequality of (22), because $u_{\mathrm{0}},v_{\mathrm{0}}$ are members of $A_V$, so we used the uniform estimates of solutions in Lemma 4 and Lemma 6 to obtain the desired result.

Similar with (22), for the second term on the right-hand side of (21), we have

$-\mathrm{2}b(P_H(\left|u(t)\right|\eta (t)),A\eta (t))\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{b}^{\mathrm{2}}{C}_{\mathrm{1}}^{\mathrm{2}}}{\gamma }\Vert \nabla u(t)\Vert \Vert \mathrm{\Delta }u(t)\Vert {\Vert \eta (t)\Vert }^{\mathrm{2}}\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\Vert \eta (t)\Vert }^{\mathrm{2}}$(23)

For the third term on the right-hand side of (21), we get

$\begin{array}{l}c(\beta +\mathrm{1})(P_H({\left|u(t)\right|}^{\beta }w(t)),A\eta (t))\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{c}^{\mathrm{2}}{(\beta +\mathrm{1})}^{\mathrm{2}}}{\mathrm{4}\gamma }{{\int }_{\mathrm{\Omega }}\left|u(t)\right|}^{\mathrm{2}\beta }{\left|w(t)\right|}^{\mathrm{2}}\mathrm{d}x\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{c}^{\mathrm{2}}{(\beta +\mathrm{1})}^{\mathrm{2}}}{\mathrm{4}\gamma }{\Vert u(t)\Vert }_{\mathrm{\infty }}^{\mathrm{2}\beta }{\Vert w(t)\Vert }^{\mathrm{2}}\\ \le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{c}^{\mathrm{2}}{(\beta +\mathrm{1})}^{\mathrm{2}}{C}_{\mathrm{1}}^{\mathrm{2}\beta }}{\mathrm{4}\gamma }{\Vert \nabla u(t)\Vert }^{\beta }{\Vert \mathrm{\Delta }u(t)\Vert }^{\beta }{\Vert w(t)\Vert }^{\mathrm{2}}\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\Vert w(t)\Vert }^{\mathrm{2}}\end{array}$(24)

Similar with (24), for the fourth term on the right-hand side of (21), we have

$-c(\beta +\mathrm{1})(P_H({\left|u(t)\right|}^{\beta }\eta (t)),A\eta (t))\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}{c}^{\mathrm{2}}{(\beta +\mathrm{1})}^{\mathrm{2}}{C}_{\mathrm{1}}^{\mathrm{2}\beta }}{\mathrm{4}\gamma }{\Vert \nabla u(t)\Vert }^{\beta }\cdot {\Vert \mathrm{\Delta }u(t)\Vert }^{\beta }{\Vert \eta (t)\Vert }^{\mathrm{2}}\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\Vert \eta (t)\Vert }^{\mathrm{2}}$(25)

For the fifth term on the right-hand side of (21), we have

$\begin{array}{l}-(P_H(b\left|u(t)\right|u(t)-b\left|v(t)\right|v(t)+c{\left|u(t)\right|}^{\beta }u(t)-c{\left|v(t)\right|}^{\beta }v(t)),A\eta (t))\\ \le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{2}\gamma }{{\int }_{\mathrm{\Omega }}\left|b\left|u(t)\right|u(t)-b\left|v(t)\right|v(t)\right|}^{\mathrm{2}}\mathrm{d}x+\frac{\mathrm{5}}{\mathrm{2}\gamma }{{\int }_{\mathrm{\Omega }}\left|c{\left|u(t)\right|}^{\beta }u(t)-c{\left|v(t)\right|}^{\beta }v(t)\right|}^{\mathrm{2}}\mathrm{d}x\end{array}$

$\begin{array}{l}\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\int }_{\mathrm{\Omega }}{(\left|u(t)\right|\left|w(t)\right|+\left|\left|u\right|-\left|v\right|\right|\left|v\right|)}^{\mathrm{2}}\mathrm{d}x+C{{\int }_{\mathrm{\Omega }}\left({\left|u(t)\right|}^{\beta }\left|w(t)\right|+\left|{\left|u(t)\right|}^{\beta }-{\left|v(t)\right|}^{\beta }\right|\left|v(t)\right|\right)}^{\mathrm{2}}\mathrm{d}x\\ \le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{{\int }_{\mathrm{\Omega }}\left|u(t)\right|}^{\mathrm{2}}{\left|w(t)\right|}^{\mathrm{2}}\mathrm{d}x+C{{\int }_{\mathrm{\Omega }}\left|v(t)\right|}^{\mathrm{2}}{\left|w(t)\right|}^{\mathrm{2}}\mathrm{d}x+C{{\int }_{\mathrm{\Omega }}\left|u(t)\right|}^{\mathrm{2}\beta }{\left|w(t)\right|}^{\mathrm{2}}\mathrm{d}x\\ +C{{\int }_{\mathrm{\Omega }}\left|{\left|u(t)\right|}^{\beta -\mathrm{1}}+{\left|v(t)\right|}^{\beta -\mathrm{1}}\right|}^{\mathrm{2}}{\left|v(t)\right|}^{\mathrm{2}}{\left|w(t)\right|}^{\mathrm{2}}\mathrm{d}x\end{array}$

$\begin{array}{l}\le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\Vert u(t)\Vert }_{\mathrm{4}}^{\mathrm{2}}{\Vert w(t)\Vert }_{\mathrm{4}}^{\mathrm{2}}+C{\Vert v(t)\Vert }_{\mathrm{4}}^{\mathrm{2}}{\Vert w(t)\Vert }_{\mathrm{4}}^{\mathrm{2}}+C{\Vert u(t)\Vert }_{\mathrm{3}\beta }^{\mathrm{2}\beta }{\Vert w(t)\Vert }_{\mathrm{6}}^{\mathrm{2}}+C({\Vert u(t)\Vert }_{\mathrm{6}(\beta -\mathrm{1})}^{\mathrm{2}(\beta -\mathrm{1})}+{\Vert v(t)\Vert }_{\mathrm{6}(\beta -\mathrm{1})}^{\mathrm{2}(\beta -\mathrm{1})}){\Vert v(t)\Vert }_{\mathrm{6}}^{\mathrm{2}}{\Vert w(t)\Vert }_{\mathrm{6}}^{\mathrm{2}}\\ \le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\Vert \nabla u(t)\Vert }^{\mathrm{2}}{\Vert \nabla w(t)\Vert }^{\mathrm{2}}+C{\Vert \nabla v(t)\Vert }^{\mathrm{2}}{\Vert \nabla w(t)\Vert }^{\mathrm{2}}+C{\Vert u(t)\Vert }_{\beta +\mathrm{2}}^{\frac{\mathrm{2}{(\beta +\mathrm{2})}^{\mathrm{2}}}{\beta +\mathrm{8}}}{\Vert \mathrm{\Delta }u(t)\Vert }^{\frac{\mathrm{8}(\beta -\mathrm{1})}{\beta +\mathrm{8}}}{\Vert \nabla w(t)\Vert }^{\mathrm{2}}\\ +C({\Vert u(t)\Vert }_{\beta +\mathrm{2}}^{\frac{\mathrm{2}({\beta }^{\mathrm{2}}+\mathrm{2}\beta )}{\beta +\mathrm{8}}}{\Vert \mathrm{\Delta }u(t)\Vert }^{\frac{\mathrm{10}\beta -\mathrm{16}}{\beta +\mathrm{8}}}+{\Vert v(t)\Vert }_{\beta +\mathrm{2}}^{\frac{\mathrm{2}({\beta }^{\mathrm{2}}+\mathrm{2}\beta )}{\beta +\mathrm{8}}}{\Vert \mathrm{\Delta }v(t)\Vert }^{\frac{\mathrm{10}\beta -\mathrm{16}}{\beta +\mathrm{8}}}){\Vert \nabla v(t)\Vert }^{\mathrm{2}}{\Vert \nabla w(t)\Vert }^{\mathrm{2}}\\ \le \frac{\gamma }{\mathrm{5}}{\Vert A\eta (t)\Vert }^{\mathrm{2}}+C{\Vert \nabla w(t)\Vert }^{\mathrm{2}}\end{array}$(26)

In inequality (26), we used $C_q$-inequality and the following Gagliardo-Nirenberg's inequality:

${\Vert u(t)\Vert }_{\mathrm{3}\beta }^{\mathrm{2}\beta }\le C{\Vert u(t)\Vert }_{\beta +\mathrm{2}}^{\frac{\mathrm{2}{(\beta +\mathrm{2})}^{\mathrm{2}}}{\beta +\mathrm{8}}}{\Vert \mathrm{\Delta }u(t)\Vert }^{\frac{\mathrm{8}(\beta -\mathrm{1})}{\beta +\mathrm{8}}},{\Vert u(t)\Vert }_{\mathrm{6}(\beta -\mathrm{1})}^{\mathrm{2}(\beta -\mathrm{1})}\le C{\Vert u(t)\Vert }_{\beta +\mathrm{2}}^{\frac{\mathrm{2}({\beta }^{\mathrm{2}}+\mathrm{2}\beta )}{\beta +\mathrm{8}}}{\Vert \mathrm{\Delta }u(t)\Vert }^{\frac{\mathrm{10}\beta -\mathrm{16}}{\beta +\mathrm{8}}}$

Combining (21)-(26), we find

$\frac{\mathrm{d}}{\mathrm{d}t}{\Vert \nabla \eta (t)\Vert }^{\mathrm{2}}+\mathrm{2}a{\Vert \nabla \eta (t)\Vert }^{\mathrm{2}}\le C{\Vert w(t)\Vert }^{\mathrm{2}}+C{\Vert \eta (t)\Vert }^{\mathrm{2}}+C{\Vert \nabla w(t)\Vert }^{\mathrm{2}}$(27)

Taking inner product with $Aw$ in $H$ to the first equation of (19), we have

$\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\Vert \nabla w(t)\Vert }^{\mathrm{2}}+\gamma {\Vert Aw(t)\Vert }^{\mathrm{2}}+a{\Vert \nabla w(t)\Vert }^{\mathrm{2}}+b(P_H(\left|u(t)\right|u(t)-\left|v(t)\right|v(t)),Aw(t))+c(P_H({\left|u(t)\right|}^{\beta }u(t)-{\left|v(t)\right|}^{\beta }v(t)),Aw(t))=\mathrm{0}$(28)

where

$\left|b(P_H(\left|u(t)\right|u(t)-\left|v(t)\right|v(t)),Aw(t))\right|\le \frac{\gamma }{\mathrm{2}}{\Vert Aw(t)\Vert }^{\mathrm{2}}+\frac{b^{\mathrm{2}}}{\mathrm{2}\gamma }{\Vert \left|u(t)\right|u(t)-\left|v(t)\right|v(t)\Vert }^{\mathrm{2}}$(29)

and

$\left|c(P_H({\left|u(t)\right|}^{\beta }u(t)-{\left|v(t)\right|}^{\beta }v(t)),Aw(t))\right|\le \frac{\gamma }{\mathrm{2}}{\Vert Aw(t)\Vert }^{\mathrm{2}}+\frac{c^{\mathrm{2}}}{\mathrm{2}\gamma }{\Vert {\left|u(t)\right|}^{\beta }u(t)-{\left|v(t)\right|}^{\beta }v(t)\Vert }^{\mathrm{2}}$(30)

For the second term on the right-hand side of (29), from inequality (26), we have

$\frac{b^{\mathrm{2}}}{\mathrm{2}\gamma }{{\int }_{\mathrm{\Omega }}\left|\left|u(t)\right|u(t)-\left|v(t)\right|v(t)\right|}^{\mathrm{2}}\mathrm{d}x\le C{\Vert \nabla u(t)\Vert }^{\mathrm{2}}{\Vert \nabla w(t)\Vert }^{\mathrm{2}}+C{\Vert \nabla v(t)\Vert }^{\mathrm{2}}{\Vert \nabla w(t)\Vert }^{\mathrm{2}}\le C{\Vert \nabla w(t)\Vert }^{\mathrm{2}}$(31)