Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Abstract

This paper deals with the asymptotic behavior of a class of second-order evolution equations with dispersive and dissipative terms’ critical nonlinearity in locally uniform spaces. First of all, we prove the global well-posedness of solutions to the evolution equations in the locally uniform spaces $H^{1}_{\textrm{lu}}({{\mathbb {R}}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$ and define a strong continuous analytic semigroup. Secondly, the existence of the $(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N) )$-global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appear to be optimal and the existence of a bounded subset(in $H^{2}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{2}_{\textrm{lu}}({\mathbb {R}}^N$)), which attracts exponentially every initial $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$-bounded set with respect to the $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$-norm.

1 Introduction

In this paper, we investigate the long-time behavior of the solutions for the following second-order evolution equations with dispersive and dissipative terms in locally uniform spaces:

$$\begin{aligned} u_{tt}-\Delta u-\Delta u_t-\beta \Delta u_{tt}+\alpha u_t+\lambda u +f(u)=g(x), \quad \text {in} \ {\mathbb {R}}^N\times {\mathbb {R}}^+,\nonumber \\ \end{aligned}$$

(1.1)

with the initial data

$$\begin{aligned} u(x,0)=u_{0},\quad u_t(x,0)=u_{1},\qquad x\in {\mathbb {R}}^N, \end{aligned}$$

(1.2)

where $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N)$ with $N\ge 3.$ The nonlinearity $f(s)\in {\mathcal {C}}^1({\mathbb {R}})$ satisfies the following conditions: Dissipative condition

$$\begin{aligned}{} & {} \liminf _{| s|\rightarrow \infty }\frac{F(s)}{s^2}\ge 0, \quad \text {where}\ F(s)=\int _0^s f(r)\textrm{d}r, \end{aligned}$$

(1.3)

$$\begin{aligned}{} & {} \liminf _{| s|\rightarrow \infty }\frac{sf(s)-\alpha F(s)}{s^2}\ge 0,\quad \text {where}\ \alpha >0. \end{aligned}$$

(1.4)

Growth condition

$$\begin{aligned} | f(s)-f(h)|{} & {} \le \beta | s-h|(1+| s|^q+| h|^q),\quad \forall ~s,h\in {\mathbb {R}},\nonumber \\{} & {} \text {where}\ \beta >0,~0\le q\le \frac{4}{N-2}. \end{aligned}$$

(1.5)

Equation (1.1) is a special form of the so-called improved Boussinesq equation (see [5, 19,20,21, 26]) with damped term $-\Delta u_t,$ which was used to describe ion-sound waves in plasma, e.g., see [20, 21], and also known to represent other sorts of ‘propagation problems’ of, for example, lengthways waves in nonlinear elastic rods and ion-sonic waves of space transformations by a weak nonlinear effect (see [5, 14]).

In bounded domains, there is a vast literature concerning the attractors for the second-order evolution equations with dispersive and dissipative terms equations. For instance, in [27, 28], Xie and Zhong investigated the existence of global attractors with critical exponential growth nonlinearity using the new method named “Condition C”. Carvalho and Cholewa in [11] presented systematic results including the existence uniqueness and long-time behavior by using the semigroup approach. Sun et al. in [24] studied the asymptotic regularity of the solutions and obtained the existence of exponential attractors. For the (nonautonomous) semi-linear second-order evolution (1.1) with memory terms, Zhang et al. in [32] constructed the existence of robust family of exponential attractors, while the nonlinearity is critical. In our previous work [33], we showed the existence of pullback attractors in the Banach spaces for the multivalued process generated by a class of second-order nonautonomous evolution equations with hereditary characteristics and ill-posedness.

On unbounded domain, up to now, there are few results. Only Jones and Wang in [18] applied the cutoff method and a decomposition trick to obtain the existence of random attractor for the stochastic second-order evolution equations (1.1) with subcritical nonlinearity.

To our best knowledge, for critical nonlinearity, the long-time dynamics for Eq. (1.1) on unbounded domain have not been considered by any predecessors. There are some barriers encountered. On the one hand, the Sobolev embeddings are not compact on unbounded domains, and hence the asymptotic compactness of solutions cannot be obtained by simply using Sobolev embeddings and regularity of solutions. On the other hand, the number $q+1=\frac{N+2}{N-2}$ in (1.5) is called a critical exponent, since the nonlinearity f is not compact even in the bounded case, and hence the methods for subcritical nonlinearity cannot be used to derive the asymptotic compactness for our problem. Thirdly, Eq. (1.1) contains the term $-\Delta {u_{tt}};$ if the initial data $z(0)=(u(0), u_t(0))$ belongs to $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),$ then the solution $z(t)=(u(t), u_t(t))$ is always in $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$ and has no higher regularity, which will cause some difficulties.

The main contributions of this paper are that:

• (i) We overcome the above difficulties (less regularity; lack of compactness; the equation itself), establish the well-posedness (Theorem 3.1), and prove the existence of bi-space global attractors for the second-order evolution equations with dispersive and dissipative terms Eq. (1.1) on ${\mathbb {R}}^N$ (Theorem 4.9).

• (ii) We obtain the asymptotic regularity of solutions on ${\mathbb {R}}^N,$ which appears to be optimal (Theorem 5.8). To our best knowledge, this is the first time to obtain the regularity for Eq. (1.1) on unbounded domain with both subcritical and critical nonlinearity, and maybe it is a basis for further considering the asymptotic behavior of the solutions.

The presentation of this paper is follows: In Sect. 2, we recall some basic definitions about the locally uniform spaces and iterate some definitions and abstract results concerning the global attractor. In Sect. 3, we prove the existence of global attractors for the second-order evolution equations with dispersive and dissipative terms in locally uniform spaces, and the asymptotic regularity of the solution will be established in Sect. 4.

2 Preliminaries

In this section, we first recall some basic definitions about the locally uniform spaces.

Following [1,2,3, 7, 8, 22, 29], we consider a strictly positive integrable weighted function $\rho :{\mathbb {R}}^N\rightarrow (0,\infty )$: for $1\le p<\infty ,$ setting

$$\begin{aligned} L^p_\rho ({\mathbb {R}}^N)=\left\{ \varphi \in L^p_{\textrm{loc}}({\mathbb {R}}^N):\Vert \varphi \Vert _{L^p_\rho ({\mathbb {R}}^N)} =\left( \int _{{\mathbb {R}}^N}\rho (x)| \varphi (x)|^p {\textrm{d}}x\right) ^{\frac{1}{p}}<\infty \right\} , \end{aligned}$$

let $\tau _y\rho (x)=\rho _y(x)=\rho (x-y),$ $y\in {\mathbb {R}}^N,$ and consider the locally uniform spaces

$$\begin{aligned}{} & {} L^p_{\textrm{lu}}({\mathbb {R}}^N)=\left\{ \varphi \in L^p_{\textrm{loc}}({\mathbb {R}}^N):\Vert \varphi \Vert _{L^p_{\textrm{lu}}({\mathbb {R}}^N)}=\sup _{y\in {\mathbb {R}}^N}\Vert \varphi \Vert _{L^p_{\rho _y}({\mathbb {R}}^N)}<\infty \right\} ,\\{} & {} \dot{L}^p_{\textrm{lu}}({\mathbb {R}}^N)=\{\varphi \in L^p_{\textrm{lu}}({\mathbb {R}}^N):\Vert \tau _y\varphi -\varphi \Vert _{L^p_{\textrm{lu}}({\mathbb {R}}^N)}\rightarrow 0\ \text {as}\ | y|\rightarrow 0\}, \end{aligned}$$

where $\dot{L}^p_{\textrm{lu}}({\mathbb {R}}^N)$ is the closed subspace of $L^p_{\textrm{lu}}({\mathbb {R}}^N)$ consisting of all its elements that are translation continuous. The locally uniform Sobolev spaces $W^{m,p}_{\textrm{lu}}({\mathbb {R}}^N)$ and $\dot{W}^{m,p}_{\textrm{lu}}({\mathbb {R}}^N)$ are defined, respectively, by $L^p_{\textrm{lu}}({\mathbb {R}}^N)$ and $\dot{L}^p_{\textrm{lu}}({\mathbb {R}}^N)$ in a way similar to the standard $W^{m,p}_{\textrm{lu}}({\mathbb {R}}^N).$

We consider strictly positive integrable weighted functions $\rho \in {\mathcal {C}}^2({\mathbb {R}}^N)$ satisfying

$$\begin{aligned} \left| \frac{\partial \rho }{\partial x_j}(x)\right| \le \rho _0\rho (x),~\left| \frac{\partial ^2 \rho }{\partial x_j\partial x_k}(x)\right| \le \rho _1\rho (x),\quad \forall x\in {\mathbb {R}}^N,j,k=1,2,\ldots ,N,\nonumber \\ \end{aligned}$$

(2.1)

with certain positive constants $\rho _0,$ $\rho _1.$ In this paper, we consider the exemplary weighted functions

$$\begin{aligned} \rho (x)=(1+\epsilon | x|^2)^{-s},\quad \text {with} \ s>\frac{N}{2},\ \epsilon >0. \end{aligned}$$

(2.2)

Obviously, $\rho \in {\mathcal {C}}^2({\mathbb {R}}^N),$ then one can obtain the estimates that $| \nabla \rho |\le c_1\sqrt{\epsilon }\rho$ and $| \Delta \rho |\le \epsilon c_2\rho .$

Now, we recall the uniform space $W^{s,p}_U({\mathbb {R}}^N),$ $s\in {\mathbb {R}}^+\cup \{0\},$ and the Banach space consisting of all $\phi \in W^{s,p}_{\textrm{loc}}({\mathbb {R}}^N)$ such that

$$\begin{aligned} \Vert \phi \Vert _{W^{s,p}_U({\mathbb {R}}^N)}=\sup _{y\in {\mathbb {R}}^N}\Vert \phi \Vert _{W^{s,p}_U(B(y,1))}<\infty , \end{aligned}$$

(2.3)

where $B(y,1)=\{x\in {\mathbb {R}}^N:| x-y|\le 1\}.$ In addition, the following two norms are equivalent: there exist $C_1,$ $C_2$ such that for all $u\in L^p_{\textrm{lu}},$

$$\begin{aligned} \Vert u\Vert ^p_{L^p_{\textrm{lu}}}{} & {} =\sup _{y\in {\mathbb {R}}^N}\int _{{\mathbb {R}}^N}\rho (x-y)| u(x)|^p \textrm{d}x\\{} & {} \le C_1 \sup _{y\in {\mathbb {R}}^N}\int _{B(y,1)}| u(x)|^p \textrm{d}x\le C_2\Vert u\Vert ^p_{L^p_{\textrm{lu}}}. \end{aligned}$$

Note that for $k\in {\mathbb {N}}\cup \{0\},$ uniform space $W^{k,p}_U({\mathbb {R}}^N)$ and locally uniform space $W^{k,p}_{\textrm{lu}}({\mathbb {R}}^N)$ coincide algebraically and topologically when the weighted function $\rho$ satisfies (2.1). Furthermore, by intermediate spaces, we know that the same holds for $W^{s,p}_U({\mathbb {R}}^N)$ and $W^{s,p}_{\textrm{lu}}({\mathbb {R}}^N)$ with $s\in {\mathbb {R}}^+\cup \{0\},$ and we will use this equivalence frequently in this paper.

In addition, we need the following embedding lemma, interpolation inequalities in the weighted spaces and locally uniform space.

Lemma 2.1

[1]

• (i) If $s_1\ge s_2\ge 0,$ $1<p_1\le p_2<\infty$ and $s_1-\frac{N}{p_1}\ge s_2-\frac{N}{p_2},$ then

  $$\begin{aligned} W_U^{s_1,p_1}({\mathbb {R}}^N)\hookrightarrow W_U^{s_2,p_2}({\mathbb {R}}^N) \end{aligned}$$

  is continuous.

• (ii) If $\rho$ satisfies (2.1), then the inclusion

  $$\begin{aligned} W_U^{s_1,p_1}({\mathbb {R}}^N)\hookrightarrow W_\rho ^{s_2,p_2}({\mathbb {R}}^N), \end{aligned}$$

  provided that $s_2\in {\mathbb {N}},$ $s_1>s_2,$ $1<p_1\le p_2<\infty$ and $s_1-\frac{N}{p_1}> s_2-\frac{N}{p_2}.$

Lemma 2.2

[1] For any $p\in [2,\frac{2N}{N-2}]$ and $\theta \in [0,1],$ we have

$$\begin{aligned} \Vert \varphi \Vert _{L^p_\rho }\le C\Vert \varphi \Vert ^\theta _{H^1_{\textrm{lu}}}\Vert \varphi \Vert ^{1-\theta }_{L^r_\rho }, \end{aligned}$$

and

$$\begin{aligned} \Vert \varphi \Vert _{L^p_\rho }\le C\Vert \varphi \Vert ^\theta _{H^1_{\rho }}\Vert \varphi \Vert ^{1-\theta }_{L^r_{\textrm{lu}}}, \end{aligned}$$

where $\frac{1}{p}\le \frac{\theta }{2}+\frac{1-\theta }{r}$ and $-\frac{N}{p}\le \theta (1-\frac{N}{2})-(1-\theta )\frac{N}{r}.$

Lemma 2.3

[31] there exist $C_1,$ $C_2$ such that for all $u\in L^p_{\rho }$ $(1\le p< \infty ),$

$$\begin{aligned} C_1\int _{{\mathbb {R}}^N}\rho (x)| u(x)|^p \textrm{d}x\le \int _{ {\mathbb {R}}^N}\rho (y)\int _{B(y,1)}| u(x)|^p \textrm{d}x\textrm{d}y\le C_2\int _{{\mathbb {R}}^N}\rho (x)| u(x)|^p \textrm{d}x. \end{aligned}$$

Next, we iterate some definitions and abstract results concerning the global attractor, which are necessary to obtain our main results; we refer to [4, 6, 9, 16, 22, 23, 25] for more details.

Definition 2.1

A set ${\mathcal {A}}\subset X,$ which is invariant, closed in X,  compact in Z and attracts the bounded subsets of X in the topology of Z,  is called an (X, Z)-global attractor.

Definition 2.2

Let $\{S(t)\}_{t\ge 0}$ be a semigroup on Banach space X. A set $B_0\subset Z,$ satisfying that, for any bounded subset $B\subset X,$ there is a $T = T(B),$ such that $S(t)B\subset B_0,$ for any $t\ge T,$ is called an (X, Z)-bounded absorbing set.

Definition 2.3

Let $\{S(t)\}_{t\ge 0}$ be a semigroup on Banach space X. $\{S(t)\}_{t\ge 0}$ is called (X, Z)-asymptotically compact, if for any bounded (in X) sequence $\{x_n\}_{n=1}^\infty \subset X$ and $t_n\ge 0,$ $t_n\rightarrow \infty$ as $n\rightarrow \infty ,$ $\{S(t_n)x_n\}_{n=1}^\infty$ has a convergence subsequence with respect to the topology of Z.

With the usual notation, hereafter let |u|,  $|\cdot |_p,$ $\Vert \cdot \Vert _{\dot{W}^{m,p}_{\textrm{lu}}},$ $\Vert \cdot \Vert _{W^{m,p}_{\textrm{lu}}},$ $\Vert \cdot \Vert _{W^{m,p}_{\rho }}$ and $\Vert \cdot \Vert _{W^{m,p}}$ be the norm of $L^2({\mathbb {R}}^N),$ $L^p({\mathbb {R}}^N),$ $\dot{W}^{m,p}_{\textrm{lu}}({\mathbb {R}}^N),$ $W^{m,p}_{\textrm{lu}}({\mathbb {R}}^N),$ $W^{m,p}_{\rho }({\mathbb {R}}^N)$ and $W^{m,p}({\mathbb {R}}^N),$ respectively. Also, let $\langle \cdot ,\cdot \rangle$ be the usual inner product in $L^2({\mathbb {R}}^N).$ Let C be an arbitrary positive constant, which may be different from line to line and even in the same line. For convenience, without loss of generality, we always assume $\alpha =\beta =\lambda =1$ hereafter.

3 Global Well-Posedness

In this section, we will investigate the well-posedness of system (1.1)–(1.2).

Theorem 3.1

(Global well-posedness) Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N).$ Then for any $T>0$ and $(u_0,u_1)\in H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),$ there is a unique solution $(u(t),u_t(t))$ of Eqs. (1.1) and (1.2) such that

$$\begin{aligned} u(t)\in {\mathcal {C}}([0,T];H^{1}_{\textrm{lu}}({\mathbb {R}}^N)), u_t(t) \in {\mathcal {C}}([0,T];H^{1}_{\textrm{lu}}({\mathbb {R}}^N)). \end{aligned}$$

Moreover,  the solution continuously depends on the initial data.

Proof

We divide the proof into three steps:

Step 1 Local well-posedness

Setting $v=(I-\Delta )u$ and $v_t=w,$ we can rewrite Eq. (1.1) into the following system:

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\left( \begin{array}{c} v\\ w \end{array}\right) +\left( \begin{array}{cc} 0 &{} -I \\ I &{} I \end{array}\right) \left( \begin{array}{c} v\\ w \end{array}\right) ={\mathcal {F}}\left( \begin{array}{c} v\\ w \end{array}\right) ,\quad t>0, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {F}}\left( \begin{array}{c} v\\ w \end{array}\right) =\left( \begin{array}{c} 0\\ f\circ ((I+A)^{-1}v)+g(x) \end{array}\right) . \end{aligned}$$

By the growth condition (1.5), $f(\cdot )$ is local Lipschitz in $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),$ which is equivalent to $(0, f\circ ((I+A)^{-1}v)+g(x))^T.$ The abstract semigroup theory about local well-posedness (e.g., see [3, 10, 11, 23, 25, 26, 29]) of an abstract parabolic equation leads to a local solution to system (1.1)–(1.2).

Step 2 Global existence

By the a priori estimates given in Lemma 4.1 below, we infer

$$\begin{aligned} \Vert u(t)\Vert ^2_{H_{\textrm{lu}}^1}+\Vert u_t(t)\Vert ^2_{H_{\textrm{lu}}^1}\le Ce^{-\nu t}(\Vert u(0)\Vert ^2_{H_{\textrm{lu}}^1}+\Vert u_t(0)\Vert ^2_{H_{\textrm{lu}}^1})+ C\int _{{\mathbb {R}}^N}\rho _y(| g|^2+1). \end{aligned}$$

This implies that for each local solution $(u(t),u_t(t))$ of system (1.1)–(1.2) corresponding to initial data $(u_0,u_1)\in H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),$ its $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$-norm cannot blow up at finite time, which implies the global existence of solutions.

Step 3 Lipschitz continuity

Let $u^1(t),$ $u^2(t)$ be two solutions of system (1.1)–(1.2) corresponding to the initial data $(u^1_0,u^1_1),(u^2_0,u^2_1)\in H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$ and denote $z(t)=u^1(t)-u^2(t),$ then z(t) satisfies

$$\begin{aligned} z_{tt}-\Delta z-\Delta z_t-\Delta z_{tt}+z_t+ z +f(u^1)-f(u^2)=0. \end{aligned}$$

(3.1)

We set $m=z_t+\eta z(0<\eta \ll 1)$ and rewrite the Eq. (3.1) as follows:

$$\begin{aligned}{} & {} m_t+(1-\eta )m+(1-\eta +\eta ^2)z-(1-\eta +\eta ^2)\Delta z\nonumber \\{} & {} \quad -(1-\eta )\Delta m-\Delta m_t +f(u^1)-f(u^2)=0. \end{aligned}$$

(3.2)

Multiplying (3.2) by $\rho _y m,$ we infer

$$\begin{aligned}{} & {} \langle m_{t},\rho _ym\rangle {+}(1{-}\eta )\langle m,\rho _ym\rangle {+}(1{-}\eta {+}\eta ^2) \langle z, \rho _ym\rangle {-}(1{-}\eta {+}\eta ^2)\langle \Delta {z},\rho _y m\rangle \nonumber \\{} & {} \quad -(1-\eta )\langle \Delta {m},\rho _ym\rangle -\langle \Delta m_t,\rho _ym\rangle +\langle f(u^1)-f(u^2),\rho _ym\rangle =0.\nonumber \\ \end{aligned}$$

(3.3)

Next, we deal with each term of (3.3) one by one as follows:

$$\begin{aligned}{} & {} \langle m_{t},\rho _ym\rangle +(1-\eta )\langle m,\rho _ym\rangle =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y | m|^2+(1-\eta )\int _{{\mathbb {R}}^N}\rho _y | m|^2, \nonumber \\ \end{aligned}$$

(3.4)

$$\begin{aligned}{} & {} \langle z,\rho _ym\rangle =\langle z,\rho _y(z_{t}+\eta z)\rangle =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y | z|^2+\eta \int _{{\mathbb {R}}^N}\rho _y | z|^2, \end{aligned}$$

(3.5)

$$\begin{aligned}{} & {} \langle -\Delta {z},\rho _ym\rangle =\langle -\Delta {z},\rho _y(z_{t}+\eta z)\rangle =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y |\nabla z|^2+\eta \int _{{\mathbb {R}}^N}\rho _y |\nabla z|^2\nonumber \\{} & {} \quad \qquad \qquad \qquad +\int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z_{t}+\eta \int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z, \end{aligned}$$

(3.6)

$$\begin{aligned}{} & {} \langle -\Delta {m},\rho _ym\rangle =\int _{{\mathbb {R}}^N}\rho _y |\nabla m|^2+\int _{{\mathbb {R}}^N}\nabla m\nabla \rho _y m, \end{aligned}$$

(3.7)

$$\begin{aligned}{} & {} \langle -\Delta {m_{t}},\rho _ym\rangle =\frac{1}{2}\frac{\textrm{d}}{ \textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y |\nabla m|^2+\int _{{\mathbb {R}}^N}\nabla m_{t}\nabla \rho _ym. \end{aligned}$$

(3.8)

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z_{t}{+}\eta \int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z{\le } C\sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(|\nabla z|^2 {+}| z_{t}|^2{+} | z|^2 ), \end{aligned}$$

(3.9)

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^N}\nabla m_{t}\nabla \rho _ym\le C\sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(|\nabla m_t|^2 + | m|^2 ), \end{aligned}$$

(3.10)

$$\begin{aligned}{} & {} (1-\eta )\int _{{\mathbb {R}}^N}\nabla m\nabla \rho _ym\le C\sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(|\nabla m|^2 + | m|^2 ). \end{aligned}$$

(3.11)

By the Sobolev embedding $H^1(B_1(r))\hookrightarrow L^{\frac{2N}{N-2}}(B_1(r))$ and $\dot{H}^1_{\textrm{lu}}({\mathbb {R}}^N)\hookrightarrow \dot{L}^{\frac{2N}{N-2}}({\mathbb {R}}^N),$ we have

$$\begin{aligned}{} & {} \left| \int _{{\mathbb {R}}^N}\rho _y (f(u^1)-f(u^2)) z\right| \nonumber \\{} & {} \quad \le \int _{{\mathbb {R}}^N}\rho _y(x)\left( 1+| u^1|^{\frac{4}{N-2}}+| u^2|^{\frac{4}{N-2}}\right) | z|^2 {\textrm{d}}x\nonumber \\{} & {} \quad \le \int _{{\mathbb {R}}^N}\rho _y(r) \left( \int _{B_1(r)}\left( 1+| u^1|^{\frac{N+2}{N-2}}+| u^2|^{\frac{4}{N-2}}\right) | z|^2 {\textrm{d}}x\right) {\textrm{d}}r\nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N}\rho _y(r) \left( \int _{B_1(r)}| z|^{\frac{2N}{N-2}} {\textrm{d}}x\right) ^{\frac{N-2}{N}}{\textrm{d}}r\nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N}\rho _y(r) | z|^2_{H^1(B_1(r))} {\textrm{d}}r\nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N}\rho _y (| \nabla z|^2+| z|^2) {\textrm{d}}x, \end{aligned}$$

(3.12)

and

$$\begin{aligned}{} & {} \left| \int _{{\mathbb {R}}^N}\rho _y (f(u^1)-f(u^2)) z_t\right| \nonumber \\{} & {} \quad \le \int _{{\mathbb {R}}^N}\rho _y(x) \left( 1+| u^1|^{\frac{4}{N-2}}+| u^2|^{\frac{4}{N-2}}\right) | z| | z_t| {\textrm{d}}x\nonumber \\{} & {} \quad \le \int _{{\mathbb {R}}^N}\rho _y(r) \left( \int _{B_1(r)}\left( 1+| u^1|^{\frac{N+2}{N-2}}+| u^2|^{\frac{4}{N-2}}\right) | z| | z_t| {\textrm{d}}x\right) {\textrm{d}}r\nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N}\rho _y(r) \left( \int _{B_1(r)}| z|^{\frac{2N}{N-2}} {\textrm{d}}x\right) ^{\frac{N-2}{2N}}\left( \int _{B_1(r)}| z_t|^{\frac{2N}{N-2}} {\textrm{d}}x\right) ^{\frac{N-2}{2N}}{\textrm{d}}r\nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N}\rho _y(r) | z|_{H^1(B_1(r))} | z_t|_{H^1(B_1(r))} {\textrm{d}}r\nonumber \\{} & {} \quad \le C_\eta \int _{{\mathbb {R}}^N}\rho _y (| \nabla z|^2+| z|^2) {\textrm{d}}x+\eta \int _{{\mathbb {R}}^N}\rho _y (| \nabla z_t|^2+| z_t|^2) {\textrm{d}}x. \end{aligned}$$

(3.13)

From (3.3)–(3.13), we get

$$\begin{aligned}{} & {} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\left( \int _{{\mathbb {R}}^N}\rho _y | m|^2+(1-\eta +\eta ^2)\int _{{\mathbb {R}}^N}\rho _y| z|^2\right. \nonumber \\{} & {} \qquad \left. +(1-\eta +\eta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla z|^2+\int _{{\mathbb {R}}^N}\rho _y | \nabla m|^2 \right) \nonumber \\{} & {} \qquad {+} (1-\eta )\int _{{\mathbb {R}}^N}\rho _y | m|^2{+}\eta (1{-}\eta {+}\eta ^2)\int _{{\mathbb {R}}^N}\rho _y(| z|^2+| \nabla z|^2)+\int _{{\mathbb {R}}^N}\rho _y| \nabla m|^2 \nonumber \\{} & {} \quad \le C \sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(| \nabla z|^2+| z|^2+| z_t|^2+| \nabla m_t|^2+| m|^2+| \nabla m|^2)\nonumber \\{} & {} \qquad +C \int _{{\mathbb {R}}^N}\rho _y(| \nabla z|^2+| z|^2)+\eta \int _{{\mathbb {R}}^N}\rho _y(| \nabla z_t|^2+| z_t|^2). \end{aligned}$$

(3.14)

In particular, we infer

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y (| m|^2+| \nabla m|^2+| z|^2+| \nabla z|^2) \nonumber \\{} & {} \qquad \le C\int _{{\mathbb {R}}^N}\rho _y (| m|^2+| \nabla m|^2+| z|^2+| \nabla z|^2). \end{aligned}$$

(3.15)

By the Gronwall Lemma, for any $T\ge 0,$ we get

$$\begin{aligned} \sup _{t\in [0,T]}(\Vert z(t)\Vert ^2_{H^{1}_{\textrm{lu}}}+\Vert z_t(t)\Vert ^2_{H^{1}_{\textrm{lu}}}) \le {\textrm{e}}^{CT}(\Vert z(0)\Vert ^2_{H^{1}_{\textrm{lu}}}+\Vert z_t(0)\Vert ^2_{H^{1}_{\textrm{lu}}}). \end{aligned}$$

(3.16)

This completes the proof. $\square$

Remark 3.2

Theorem 3.1 implies that the solution of Eqs. (1.1)–(1.2) generates a ${\mathcal {C}}^0$ semigroup $\{S(t)\}_{t\ge 0}$ defined by

$$\begin{aligned} S(t):H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\rightarrow H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\\ \text {and}\quad S(t):(u_0,u_1)\mapsto (u(t),u_t(t)). \end{aligned}$$

Moreover, the semigroup $\{S(t)\}_{t\ge 0}$ satisfying the Lipschitz continuity: given any $R>0$ and any two initial data $(u^1_0,u^1_1),(u^2_0,u^2_1)\in H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$ with $\Vert (u^i_0,u^i_1)\Vert _{H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)}\le R,$ $i=1,2,$ it holds that:

$$\begin{aligned}{} & {} \Vert S(t)(u^1_0,u^1_1)-S(t)(u^2_0,u^2_1)\Vert _{H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)}\\{} & {} \qquad \le {\textrm{e}}^{C_Rt}\Vert (u^1_0,u^1_1)-(u^2_0,u^2_1)\Vert _{H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)},\quad \forall ~ t\ge 0. \end{aligned}$$

4 Global Attractor

In the section, we will prove the existence of global attractor for a class of second-order evolution equations with dispersive and dissipative terms in locally uniform spaces.

4.1 Dissipation Estimates

Lemma 4.1

Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N).$ There is a positive constant $\varrho _1$ such that for any bounded subset $B\subset \dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N)\times \dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N),$ there exists a positive constant $T_1 = T_1(B)$ such that

$$\begin{aligned} \Vert u(t)\Vert _{H_{\textrm{lu}}^1}+\Vert u_t(t)\Vert _{H_{\textrm{lu}}^1}\le \varrho _1\quad \text {for all }t\ge T_1 \text { and } (u_0,u_1)\in B. \end{aligned}$$

(4.1)

Proof

We set $v=u_t+\delta u(0<\delta \ll 1)$ and rewrite Eq. (1.1) as follows

$$\begin{aligned}{} & {} v_t+(1-\delta )v+(1- \delta +\delta ^2)u-(1-\delta +\delta ^2)\Delta u\nonumber \\{} & {} \quad -(1-\delta )\Delta v-\Delta v_t +f(u)=g(x). \end{aligned}$$

(4.2)

Multiplying (4.2) by $\rho _yv,$ we infer

$$\begin{aligned}{} & {} \langle v_{t},\rho _yv\rangle +(1-\delta )\langle v,\rho _yv\rangle +(1- \delta +\delta ^2) \langle u, \rho _yv\rangle -(1-\delta +\delta ^2)\langle \Delta {u},\rho _yv\rangle \nonumber \\{} & {} \quad -(1-\delta )\langle \Delta {v},\rho _yv\rangle -\langle \Delta v_t,\rho _yv\rangle +\langle f(u),\rho _yv\rangle =\langle g(x),\rho _yv\rangle . \end{aligned}$$

(4.3)

Next, we deal with each term of (4.3) one by one as follows:

$$\begin{aligned}{} & {} \langle v_{t},\rho _yv\rangle +(1-\delta )\langle v,\rho _yv\rangle =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y | v|^2+(1-\delta )\int _{{\mathbb {R}}^N}\rho _y | v|^2, \end{aligned}$$

(4.4)

$$\begin{aligned}{} & {} \langle u,\rho _yv\rangle =\langle u,\rho _y(u_{t}+\delta u)\rangle =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y | u|^2+\delta \int _{{\mathbb {R}}^N}\rho _y | u|^2, \end{aligned}$$

(4.5)

$$\begin{aligned}{} & {} \langle -\Delta {u},\rho _yv\rangle =\langle -\Delta {u},\rho _y(u_{t}+\delta u)\rangle =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y |\nabla u|^2+\delta \int _{{\mathbb {R}}^N}\rho _y |\nabla u|^2\nonumber \\{} & {} \quad + \int _{{\mathbb {R}}^N}\nabla u\nabla \rho _y u_{t}+\delta \int _{{\mathbb {R}}^N}\nabla u\nabla \rho _y u, \end{aligned}$$

(4.6)

$$\begin{aligned}{} & {} \langle -\Delta {v},\rho _yv\rangle =\int _{{\mathbb {R}}^N}\rho _y |\nabla v|^2+\int _{{\mathbb {R}}^N}\nabla v\nabla \rho _y v, \end{aligned}$$

(4.7)

$$\begin{aligned}{} & {} \langle -\Delta {v_{t}},\rho _yv\rangle =\frac{1}{2}\frac{\textrm{d}}{ \textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y |\nabla v|^2+\int _{{\mathbb {R}}^N}\nabla v_{t}\nabla \rho _y v, \end{aligned}$$

(4.8)

$$\begin{aligned}{} & {} \langle f(u),\rho _yv\rangle =\langle f(u),\rho _y(u_{t}+\delta u)\rangle \nonumber \\{} & {} \qquad \qquad \qquad =\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _yF(u)+\delta \int _{{\mathbb {R}}^N}\rho _yf(u)u, \end{aligned}$$

(4.9)

$$\begin{aligned}{} & {} \langle g(x),\rho _y v\rangle \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v|^2+C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y| g|^2. \end{aligned}$$

(4.10)

From (4.3)–(4.10), we get

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\Bigg (\int _{{\mathbb {R}}^N}\rho _y | v|^2+(1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| u|^2+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla u|^2\nonumber \\{} & {} \qquad +\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2 +2\int _{{\mathbb {R}}^N}\rho _yF(u) \Bigg ) \nonumber \\{} & {} \qquad +2(1-\delta -\varsigma )\int _{{\mathbb {R}}^N}\rho _y | v|^2+2\delta (1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y | u|^2\nonumber \\{} & {} \qquad +2\delta (1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y |\nabla u|^2\nonumber \\{} & {} \qquad +2(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla u\nabla \rho _y u_{t} \nonumber \\{} & {} \qquad +2\delta (1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla u\nabla \rho _y u \nonumber \\{} & {} \qquad +2(1-\delta )\int _{{\mathbb {R}}^N}\rho _y |\nabla v|^2+2(1-\delta )\int _{{\mathbb {R}}^N}\nabla v\nabla \rho _y v+2\int _{{\mathbb {R}}^N}\nabla v_{t}\nabla \rho _y v\nonumber \\{} & {} \qquad +2\delta \int _{{\mathbb {R}}^N}\rho _yf(u)u \nonumber \\{} & {} \quad \le C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y| g|^2. \end{aligned}$$

(4.11)

Noting that

$$\begin{aligned}{} & {} 2(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla u\nabla \rho _y u_{t} +2\delta (1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla u\nabla \rho _y u \nonumber \\{} & {} \quad \le C\sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(|\nabla u|^2 +| u_{t}|^2+ | u|^2 ), \end{aligned}$$

(4.12)

$$\begin{aligned}{} & {} 2(1-\delta )\int _{{\mathbb {R}}^N}\nabla v\nabla \rho _y v+2\int _{{\mathbb {R}}^N}\nabla v_{t}\nabla \rho _y v \nonumber \\{} & {} \quad \le C\sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(|\nabla v|^2 +| v_{t}|^2+ | v|^2 ). \end{aligned}$$

(4.13)

By (1.3)–(1.4), we infer

$$\begin{aligned} \int _{{\mathbb {R}}^N}\rho _y f(u)u\ge c_1 \int _{{\mathbb {R}}^N}\rho _yF(u)+\mu \int _{{\mathbb {R}}^N}\rho _y| u|^2-C_\mu \int _{{\mathbb {R}}^N}\rho _y, \end{aligned}$$

(4.14)

and

$$\begin{aligned} \int _{{\mathbb {R}}^N}\rho _y F(u)\ge -c_2 \int _{{\mathbb {R}}^N}\rho _y. \end{aligned}$$

(4.15)

Substituting the estimates (4.12)–(4.15) into (4.11), and choosing $\epsilon$ and $\varsigma$ small enough, we infer that

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\Bigg (\int _{{\mathbb {R}}^N}\rho _y | v|^2+(1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| u|^2+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla u|^2\nonumber \\{} & {} \qquad +\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2 +2\int _{{\mathbb {R}}^N}\rho _yF(u) \Bigg ) \nonumber \\{} & {} \qquad +\nu \Bigg (\int _{{\mathbb {R}}^N}\rho _y | v|^2+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| u|^2+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla u|^2\nonumber \\{} & {} \qquad +\beta \int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2 +2\int _{{\mathbb {R}}^N}\rho _yF(u) \Bigg ) \nonumber \\{} & {} \quad \le C_\delta \int _{{\mathbb {R}}^N}\rho _y| g|^2+ C, \end{aligned}$$

(4.16)

where $\nu$ is a positive constant which depends on $\delta$ and $\mu .$

Denoting

$$\begin{aligned} {\mathcal {E}}_1(t)= & {} \int _{{\mathbb {R}}^N}\rho _y | v|^2+(1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| u|^2\nonumber \\{} & {} \quad +(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla u|^2+\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2 +2\int _{{\mathbb {R}}^N}\rho _yF(u),\nonumber \\ \end{aligned}$$

(4.17)

we can obtain that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} {\mathcal {E}}_1(t)+\nu {\mathcal {E}}_1(t)\le C\int _{{\mathbb {R}}^N}\rho _y(| g|^2+1). \end{aligned}$$

(4.18)

Using the Gronwall lemma, we infer

$$\begin{aligned} \mathcal { E}_1(t)\le {\textrm{e}}^{-\nu t}{\mathcal {E}}_1(0)+ C\int _{{\mathbb {R}}^N}\rho _y(| g|^2+1). \end{aligned}$$

(4.19)

Noting that ${\mathcal {E}}_1(t)\sim \Vert u(t)\Vert ^2_{H_{\textrm{lu}}^1}+\Vert u_t(t)\Vert ^2_{H_{\textrm{lu}}^1},$ this completes the proof.

Remark 4.2

Lemma 4.1 implies that the ${\mathcal {C}}^0$ semigroup $\{S(t)\}_{t\ge 0}$ has a $(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N))$-bounded absorbing set in the locally uniform space $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N).$

Lemma 4.3

Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N).$ There is a positive constant $\varrho _2$ such that for any bounded subset $B\subset \dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N)\times \dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N),$ there exists a positive constant $T_2 = T_2(B)$ such that

$$\begin{aligned} \Vert u_t(t)\Vert _{H_{\textrm{lu}}^1}{+}\Vert u_{tt}(t)\Vert _{H_{\textrm{lu}}^1}{\le }\varrho _2\quad \text {for all }t\ge T_2 \text { and } (u_0,u_1)\in B. \end{aligned}$$

(4.20)

Proof

Multiplying (4.2) by $\rho _y v_{t},$ we infer

$$\begin{aligned}{} & {} \langle v_{t},\rho _y v_{t}\rangle +(1-\delta )\langle v,\rho _y v_{t}\rangle +(1- \delta +\delta ^2) \langle u, \rho _y v_{t}\rangle \nonumber \\{} & {} \quad -(1-\delta +\delta ^2)\langle \Delta {u},\rho _y v_{t}\rangle -(1-\delta )\langle \Delta {v},\rho _y v_{t}\rangle -\langle \Delta v_t,\rho _y v_{t}\rangle \nonumber \\{} & {} \quad +\langle f(u),\rho _y v_{t}\rangle =\langle g(x),\rho _y v_{t}\rangle . \end{aligned}$$

(4.21)

Next, we deal with each term of (4.21) one by one as follows:

$$\begin{aligned}{} & {} \langle v_{t},\rho _y v_{t} \rangle +(1-\delta )\langle v,\rho _y v_{t} \rangle =\int _{{\mathbb {R}}^N}\rho _y | v_{t}|^2+\frac{1-\delta }{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y | v|^2,\nonumber \\ \end{aligned}$$

(4.22)

$$\begin{aligned}{} & {} |(1- \delta +\delta ^2) \langle u, \rho _y v_{t}\rangle | \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_{t}|^2+C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y | u |^2 \nonumber \\{} & {} \quad \qquad \qquad \qquad \qquad \qquad \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_{t}|^2 +C_{\varsigma ,\varrho _1}, \end{aligned}$$

(4.23)

$$\begin{aligned}{} & {} (1-\delta +\delta ^2)\langle -\Delta {u},\rho _yv_t\rangle \nonumber \\{} & {} \quad =(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla u\nabla \rho _y v_{t}+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla u\rho _y \nabla v_t \nonumber \\{} & {} \quad \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_{t}|^2 +\varsigma \int _{{\mathbb {R}}^N}\rho _y | \nabla v_{t}|^2 +C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y | \nabla u|^2 \nonumber \\{} & {} \quad \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_{t}|^2 +\varsigma \int _{{\mathbb {R}}^N}\rho _y | \nabla v_{t}|^2 +C_{c_1,\varsigma ,\varrho _1},\nonumber \\ \end{aligned}$$

(4.24)

$$\begin{aligned}{} & {} (1-\delta )\langle -\Delta {v},\rho _yv_t\rangle \nonumber \\{} & {} \quad =(1-\delta )\int _{{\mathbb {R}}^N}\nabla v\nabla \rho _y v_{t}+(1-\delta )\int _{{\mathbb {R}}^N}\nabla v\rho _y \nabla v_t \nonumber \\{} & {} \quad \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_{t}|^2 +C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2 +\frac{1-\delta }{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2 \nonumber \\{} & {} \quad \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_{t}|^2 +\frac{1-\delta }{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2+C_{c_1,\varsigma ,\varrho _1}, \end{aligned}$$

(4.25)

$$\begin{aligned}{} & {} -\langle \Delta v_t,\rho _y v_{t}\rangle =\int _{{\mathbb {R}}^N}\nabla v_t\nabla \rho _y v_{t}+\int _{{\mathbb {R}}^N}\rho _y | \nabla v_t| ^2, \end{aligned}$$

(4.26)

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^N}\nabla v_t\nabla \rho _y v_{t}\le C_{c_1}\sqrt{\epsilon }(| \nabla v_t| ^2+| v_t| ^2). \end{aligned}$$

(4.27)

$$\begin{aligned}{} & {} \left| \int _{{\mathbb {R}}^N}\rho _y f(u)v_t\right| \le \int _{{\mathbb {R}}^N}\rho _y(r) \left( \int _{B_1(r)}(1+| u|^{\frac{N+2}{N-2}})| v_t| {\textrm{d}}x\right) {\textrm{d}}r\nonumber \\{} & {} \quad \le \int _{{\mathbb {R}}^N}\rho _y(r) \left( \int _{B_1(r)}(1+| u|^{\frac{N+2}{N-2}})^{\frac{2N}{N+2}}\right) ^{\frac{N+2}{2N}}\left( \int _{B_1(r)}| v_t|^{\frac{2N}{N-2}} {\textrm{d}}x\right) ^{\frac{N-2}{2N}}{\textrm{d}}r\nonumber \\{} & {} \quad \le \int _{{\mathbb {R}}^N}\rho _y(r) \left( \int _{B_1(r)}\left( 1+| u|^{\frac{N+2}{N-2}}\right) ^{\frac{2N}{N+2}}\right) ^{\frac{N+2}{2N}}\left( \int _{B_1(r)}| v_t|^{\frac{2N}{N-2}} {\textrm{d}}x\right) ^{\frac{N-2}{2N}}{\textrm{d}}r\nonumber \\{} & {} \quad \le \int _{{\mathbb {R}}^N}\rho _y(r) \Vert u\Vert _{H^1(B_1(r))} \Vert v_t\Vert _{B_1(r)}{\textrm{d}}r \nonumber \\{} & {} \quad \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_t|^2+C_{\varsigma }\Vert u\Vert ^2_{H^1_{\textrm{lu}}} \nonumber \\{} & {} \quad \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_t|^2+C_{\varsigma ,\varrho _1}.\nonumber \\ \end{aligned}$$

(4.28)

$$\begin{aligned}{} & {} \langle g(x),\rho _y v_{t}\rangle \le \varsigma \int _{{\mathbb {R}}^N}\rho _y | v_t|^2+ C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y | g|^2. \end{aligned}$$

(4.29)

Substituting the estimates (4.22)–(4.29) into (4.21), and choosing $\epsilon$ and $\varsigma$ small enough, we get that

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\left( (1-\delta )\int _{{\mathbb {R}}^N}\rho _y | v|^2+(1-\delta )\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2\right) \nonumber \\{} & {} \qquad +2(1-5\varsigma -C_{c_1,\sqrt{\epsilon }})\int _{{\mathbb {R}}^N}\rho _y | v_t|^2 +2(1-\varsigma -C_{c_1,\sqrt{\epsilon }})\int _{{\mathbb {R}}^N}\rho _y | \nabla v_t|^2 \nonumber \\{} & {} \quad \le C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y | g|^2+C_{c_1,\varsigma ,\varrho _1}. \end{aligned}$$

(4.30)

In particular, we have

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\left( (1-\delta )\int _{{\mathbb {R}}^N}\rho _y | v|^2+(1-\delta )\int _{{\mathbb {R}}^N}\rho _y | \nabla v|^2\right) \nonumber \\{} & {} \quad \le C_{\varsigma }\int _{{\mathbb {R}}^N}\rho _y | g|^2+C_{c_1,\varsigma ,\varrho _1}. \end{aligned}$$

(4.31)

We infer that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\rho _y | v(t)|^2+\int _{{\mathbb {R}}^N}\rho _y | \nabla v(t)|^2\le C + C\int _{{\mathbb {R}}^N}\rho _y| g|^2. \end{aligned}$$

(4.32)

This completes the proof. $\square$

4.2 Decomposition of the Equations

For the nonlinear term f,  following the idea in [12, 13, 15, 29,30,31], for a ${\mathcal {C}}^1$-function $f(\cdot )$ satisfying (1.3)–(1.5), the following decomposing properties hold: there are constants $C > 0$ and $\gamma$ satisfying $0< \gamma < q + 1$ such that f can be decomposed as

$$\begin{aligned} f = f_0 + f_1 \end{aligned}$$

with $f_0,$ $f_1\in {\mathcal {C}}^1({\mathbb {R}})$ satisfying

$$\begin{aligned}{} & {} | f_0(s)|\le C(| s|+| s|^{q+1}),\quad ~\forall ~s\in {\mathbb {R}}, \end{aligned}$$

(4.33)

$$\begin{aligned}{} & {} f_0(s)s\ge 0,\quad \forall ~s\in {\mathbb {R}}, \end{aligned}$$

(4.34)

$$\begin{aligned}{} & {} |f_1(s)|\le C(1+| s|^{\gamma }),\quad \forall ~s\in {\mathbb {R}} \text { with some }\gamma <q+1, \end{aligned}$$

(4.35)

$$\begin{aligned}{} & {} \exists ~l\in {\mathbb {R}},~F_1(s)\ge -l,\quad \forall ~s\in {\mathbb {R}}, \end{aligned}$$

(4.36)

$\exists k_i\ge 1,~{\tilde{\mu }}_i\ge 0$ such that $\forall \mu _i\in (0,{\tilde{\mu }}_i],$ $\exists C_{\mu _i}\in {\mathbb {R}},$

$$\begin{aligned} k_iF_i(s)+\mu _i s^2-C_{\mu _i}\le sf_i(s),\quad \text {for all }s\in {\mathbb {R}}, \end{aligned}$$

(4.37)

where $F_i(s)=\int _0^s f_i(r){\textrm{d}}r,$ $i=1,2.$

Now, we decompose the solution into the sum

$$\begin{aligned} S(t)(u_0,u_1)=D(t)(u_0,u_1)+K(t)(u_0,u_1), \end{aligned}$$

where $(z(t), z_t(t))=D(t)(u_0,u_1)$ and $(w(t),w_t(t))=K(t)(u_0,u_1)$ solves the following equations, respectively:

$$\begin{aligned} \left\{ \begin{aligned}&\ z_{tt}-\Delta z-\Delta z_t-\Delta z_{tt}+ z_t+ z+f_0(z)=0, \\&\ z(0)=u_0,z_t(0)=u_1,\\ \end{aligned} \right. \end{aligned}$$

(4.38)

and

$$\begin{aligned} \left\{ \begin{aligned}&\ w_{tt}-\Delta w-\Delta w_t-\Delta w_{tt}+ w_t+ w+f(u)-f_0(z)=g(x), \\&\ w(0)=0,w_t(0)=0.\\ \end{aligned} \right. \end{aligned}$$

(4.39)

Note that $\{D(t)\}_{t\ge 0}$ also forms a semigroup, but $\{K(t)\}_{t\ge 0}$ may not.

4.3 A Priori Estimates

Lemma 4.4

Assume that $f_0$ satisfies (4.33)–(4.34), (4.37). Then there is a positive constant $\varrho _3$ such that for any bounded subset $B\subset \dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N)\times \dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N),$ there exists a positive constant $T_3 = T_3(B)$ such that

$$\begin{aligned} \Vert z_t(t)\Vert _{H_{\textrm{lu}}^1}^2+\Vert z_{tt}(t)\Vert _{H_{\textrm{lu}}^1}^2 \le \varrho _3,\quad \forall t\ge T_3,\ (u_0,u_1)\in B. \end{aligned}$$

(4.40)

The proof of this Lemma is a repeat of Lemma 4.3, and we omit the details.

Remark 4.5

D(t) maps the bounded set of $\dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N)\times \dot{W}^{1,2}_{\textrm{lu}}({\mathbb {R}}^N)$ to be a uniformly (w.r.t. time t) bounded set; that is, for any $(u_0,u_1)\in {H_{\textrm{lu}}^1}\times {H_{\textrm{lu}}^1},$

$$\begin{aligned} \Vert D(t)(u_0,u_1)\Vert _{H_{\textrm{lu}}^1}^2= & {} \Vert z(t)\Vert _{H_{\textrm{lu}}^1}^2+\Vert z_t(t)\Vert _{H_{\textrm{lu}}^1}^2 \nonumber \\{} & {} \le {\mathcal {Q}}(\Vert (u_0,u_1)\Vert _{_{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1}}),\quad \text {for all }t\ge 0. \end{aligned}$$

(4.41)

For the solutions $(z(t), z_t(t))=D(t)(u_0,u_1)$ of Eq. (4.38), we also need the following exponential decay result.

Lemma 4.6

Assume that $f_0$ satisfies (4.33)–(4.34), (4.37). Then there exists a positive constant $\nu$ such that for every $t\ge T_3,$

$$\begin{aligned} \Vert D(t)(u_0,u_1)\Vert _{H_{\textrm{lu}}^1}^2= \Vert z(t)\Vert _{H_{\textrm{lu}}^1}^2+\Vert z_t(t)\Vert _{H_{\textrm{lu}}^1}^2 \le {\mathcal {Q}}_1(\Vert (u_0,u_1)\Vert _{_{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1}}){\textrm{e}}^{-\nu t},\nonumber \\ \end{aligned}$$

(4.42)

where ${\mathcal {Q}}_1(\cdot )$ is an increasing function on $[0,\infty ).$

Proof

We set $q=z_t+\delta z(0<\delta \ll 1)$ and rewrite the Eq. (4.38) as follows:

$$\begin{aligned}{} & {} q_t+(1-\delta )q+(1- \delta +\delta ^2)z-(1-\delta +\delta ^2)\Delta z-(1-\delta )\Delta q-\Delta q_t \nonumber \\{} & {} \quad +f_0(z)=0. \end{aligned}$$

(4.43)

Multiplying (4.43) by $\rho _yq,$ we infer that

$$\begin{aligned}{} & {} \langle q_{t},\rho _yq\rangle +(1-\delta )\langle q,\rho _yq\rangle +(1- \delta +\delta ^2) \langle z, \rho _yq\rangle -(1-\delta +\delta ^2)\langle \Delta {z},\rho _yq\rangle \nonumber \\{} & {} \quad -(1-\delta )\langle \Delta {q},\rho _yq\rangle -\langle \Delta q_t,\rho _yq\rangle +\langle f(z),\rho _yq\rangle =0. \end{aligned}$$

(4.44)

By some standard calculations, we infer that

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\Bigg (\int _{{\mathbb {R}}^N}\rho _y | q|^2+(1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| z|^2+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla z|^2\nonumber \\{} & {} \qquad +\int _{{\mathbb {R}}^N}\rho _y | \nabla q|^2 +2\int _{{\mathbb {R}}^N}\rho _yF_0(z)\Bigg ) \nonumber \\{} & {} \qquad + 2(1-\delta )\int _{{\mathbb {R}}^N}\rho _y | q|^2+2\delta (1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y | z|^2\nonumber \\{} & {} \qquad +2\delta (1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y |\nabla z|^2 \nonumber \\{} & {} \qquad +2(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z_{t} +2\delta (1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z \nonumber \\{} & {} \qquad +2(1-\delta )\int _{{\mathbb {R}}^N}\rho _y |\nabla q|^2+2(1-\delta )\int _{{\mathbb {R}}^N}\nabla v\nabla \rho _y q\nonumber \\{} & {} \qquad +2\int _{{\mathbb {R}}^N}\nabla q_{t}\nabla \rho _y v+2\delta \int _{{\mathbb {R}}^N}\rho _yf_0(z)z \nonumber \\{} & {} \quad = 0. \end{aligned}$$

(4.45)

Noting that

$$\begin{aligned}{} & {} | 2(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z_{t} +2\delta (1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z | \nonumber \\{} & {} \quad \le C\sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(|\nabla z|^2 +| z_{t}|^2+ | z|^2 ), \end{aligned}$$

(4.46)

$$\begin{aligned}{} & {} 2(1-\delta )\int _{{\mathbb {R}}^N}\nabla z\nabla \rho _y z+2\int _{{\mathbb {R}}^N}\nabla z_{t}\nabla \rho _y z \nonumber \\{} & {} \quad \le C\sqrt{\epsilon }\int _{{\mathbb {R}}^N}\rho _y(|\nabla z|^2 +| z_{t}|^2+ | z|^2 ). \end{aligned}$$

(4.47)

By (4.33), Lemma 4.4 and Remark 4.5, we infer that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\rho _y F_0(z)\le & {} C\int _{{\mathbb {R}}^N}\rho _y (| z|^2+| z|^{\frac{2N}{N-2}}) \nonumber \\\le & {} C \Vert z\Vert ^{\frac{4}{N-2}}_{H^1_{\textrm{lu}}}\int _{{\mathbb {R}}^N}\rho _y (| z|^2+| \nabla z|^2) \nonumber \\\le & {} C_{\Vert (u_0,u_1)\Vert _{_{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1}}} \int _{{\mathbb {R}}^N}\rho _y (| z|^2+| \nabla z|^2). \end{aligned}$$

(4.48)

Note that

$$\begin{aligned} f_0(z)z\ge 0,\quad \forall ~z\in {\mathbb {R}};\qquad F_0(z)\ge 0, \quad \forall ~z\in {\mathbb {R}}. \end{aligned}$$

(4.49)

Hence, choosing $\epsilon$ small enough, we infer that

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\Bigg (\int _{{\mathbb {R}}^N}\rho _y | q|^2+(1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| z|^2+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla z|^2\nonumber \\{} & {} \quad +\int _{{\mathbb {R}}^N}\rho _y | \nabla q|^2 +2\int _{{\mathbb {R}}^N}\rho _yF_0(z)\Bigg ) \nonumber \\{} & {} \quad +\nu \Bigg (\int _{{\mathbb {R}}^N}\rho _y | q|^2+(1- \delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| z|^2+(1-\delta +\delta ^2)\int _{{\mathbb {R}}^N}\rho _y| \nabla z|^2\nonumber \\{} & {} \quad +\int _{{\mathbb {R}}^N}\rho _y | \nabla q|^2 +2\int _{{\mathbb {R}}^N}\rho _yF_0(z)\Bigg ) \le 0, \end{aligned}$$

(4.50)

where $\nu$ is a positive constant which depends on $\delta .$

Applying the Gronwall Lemma, we get

$$\begin{aligned} \int _{{\mathbb {R}}^N}\rho _y | q|^2+ & {} \int _{{\mathbb {R}}^N}\rho _y| z|^2+\int _{{\mathbb {R}}^N}\rho _y| \nabla z|^2+\int _{{\mathbb {R}}^N}\rho _y | \nabla q|^2 \nonumber \\{} & {} \le {\mathcal {Q}}_1(\Vert (u_0,u_1)\Vert _{_{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1}}){\textrm{e}}^{-\nu t}. \end{aligned}$$

(4.51)

This completes the proof. $\square$

Remark 4.7

Based on Lemmas 4.1, 4.3–4.6, the solutions $(w(t),w_t(t))=K(t)(u_0,u_1)$ of Eq. (4.39), we infer that there exists a positive constant $\varrho _4$ such that

$$\begin{aligned} \Vert w(t)\Vert _{H_{\textrm{lu}}^1}^2+\Vert w_t(t)\Vert _{H_{\textrm{lu}}^1}^2+\Vert w_{tt}(t)\Vert _{H_{\textrm{lu}}^1}^2\le \varrho _4,\quad \forall t\ge 0\ \text {and}\ (u_0,u_1)\in B.\nonumber \\ \end{aligned}$$

(4.52)

Lemma 4.8

Assume that f satisfies (1.3)–(1.5), $f_1$ satisfies (4.35)–(4.37), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N).$ Then there exists a positive constant k such that for every $t\ge 0,$

$$\begin{aligned} \Vert K(t)(u_0,u_1)\Vert _{H_{\textrm{lu}}^{1+\sigma }}^2= & {} \Vert w(t)\Vert _{H_{\textrm{lu}}^{1+\sigma }}^2+\Vert w_t(t)\Vert _{H_{\textrm{lu}}^{1+\sigma }}^2\nonumber \\{} & {} \le {\mathcal {Q}}_2(\Vert (u_0,u_1)\Vert _{_{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1}},\Vert g\Vert _{L_{\textrm{lu}}^2}){\textrm{e}}^{kt}, \end{aligned}$$

(4.53)

where ${\mathcal {Q}}_2(\cdot )$ is an increasing function on $[0,\infty ),$ $\sigma =\min \{\frac{1}{4},\frac{N+2-(N-2)\gamma }{2}\},$ where $\gamma$ is given in (4.35).

Proof

Let $\theta$ be a smooth function satisfying $0\le \theta (s)\le 1$ for $s\in [0,\infty )$ and

$$\begin{aligned} \theta (s)=1\quad \text {for}\ 0\le s\le \frac{1}{2};\qquad \theta (s)=0\quad \text {for}\ s\ge 1. \end{aligned}$$

Set $\theta _y(x)=\theta (| x-y|)$ and $A=-\Delta .$

We set $m=w_t+\delta w(0<\delta \ll 1)$ and rewrite the Eq. (4.39) as follows:

$$\begin{aligned}{} & {} m_t+(1-\delta )m+(1- \delta +\delta ^2)w-(1-\delta +\delta ^2)\Delta w -(1-\delta )\Delta m-\Delta m_t \nonumber \\{} & {} \quad +f(u)-f(z)+f_1(z)=g(x). \end{aligned}$$

(4.54)

Multiplying (4.54) by $\theta _yA^\sigma (\theta _ym),$ we infer that

$$\begin{aligned}{} & {} \langle m_{t},\theta _yA^\sigma (\theta _ym)\rangle +(1-\delta )\langle m,\theta _yA^\sigma (\theta _ym)\rangle +(1- \delta +\delta ^2) \langle w, \theta _yA^\sigma (\theta _ym)\rangle \nonumber \\{} & {} \quad -(1-\delta +\delta ^2)\langle \Delta {w},\theta _yA^\sigma (\theta _ym) \rangle -(1-\delta )\langle \Delta {m},\theta _yA^\sigma (\theta _ym)\rangle \nonumber \\{} & {} \quad -\langle \Delta m_t,\theta _yA^\sigma (\theta _ym)\rangle +\langle f(u)-f(z)+f_1(z),\theta _yA^\sigma (\theta _ym)\rangle \nonumber \\{} & {} =\langle g(x),\theta _yA^\sigma (\theta _ym)\rangle .\nonumber \\ \end{aligned}$$

(4.55)

We deal with each term above, one by one, as follows:

$$\begin{aligned}{} & {} \langle m_{t},\theta _yA^\sigma (\theta _ym)\rangle +(1-\delta )\langle m,\theta _yA^\sigma (\theta _ym)\rangle \nonumber \\{} & {} \quad = \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}| \theta _yA^{\frac{\sigma }{2}}(\theta _ym)|^2+(1-\delta )\int _{{\mathbb {R}}^N} | A^{\frac{\sigma }{2}}(\theta _ym)|^2,\nonumber \\ \end{aligned}$$

(4.56)

$$\begin{aligned}{} & {} \langle w, \theta _yA^\sigma (\theta _ym)\rangle \nonumber \\{} & {} \quad =\langle w, \theta _yA^\sigma (\theta _y (w_t+\delta w))\rangle \nonumber \\{} & {} \quad = \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N} | \theta _yA^{\frac{\sigma }{2}}(\theta _yw)|^2+\delta \int _{{\mathbb {R}}^N}| \theta _yA^{\frac{\sigma }{2}}(\theta _yw)|^2,\nonumber \\ \end{aligned}$$

(4.57)

$$\begin{aligned}{} & {} -\langle \Delta {w},\theta _yA^\sigma (\theta _ym)\rangle \nonumber \\{} & {} \quad =\langle A{w},\theta _yA^\sigma (\theta _y(w_t+\delta w))\rangle \nonumber \\{} & {} \quad =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^N}| A^{\frac{\sigma +1}{2}}(\theta _yw)|^2+2\int _{{\mathbb {R}}^N}\nabla w\nabla \theta _y A^\sigma (\theta _yw_t)+\langle \Delta \theta _y w, A^\sigma (\theta _y w_t) \rangle \nonumber \\{} & {} \qquad +\delta \int _{{\mathbb {R}}^N}| A^{\frac{\sigma +1}{2}}(\theta _yw)|^2+2\delta \int _{{\mathbb {R}}^N}\nabla w\nabla \theta _y A^\sigma (\theta _yw)+\delta \langle \Delta \theta _y w, A^\sigma (\theta _y w)\rangle ,\nonumber \\ \end{aligned}$$

(4.58)

$$\begin{aligned}{} & {} \langle -\Delta {m_{t}},\theta _yA^\sigma (\theta _ym)\rangle =\frac{1}{2}\frac{\textrm{d}}{ \textrm{d}t}\int _{{\mathbb {R}}^N}| A^{\frac{\sigma +1}{2}}(\theta _ym)|^2+2\int _{{\mathbb {R}}^N}\nabla m_{t}\nabla \theta _y A^\sigma (\theta _ym)\nonumber \\{} & {} \qquad +\langle \Delta \theta _y m_t, A^\sigma (\theta _y m_t)\rangle ,\nonumber \\ \end{aligned}$$

(4.59)

$$\begin{aligned}{} & {} -\langle \Delta {m},\theta _yA^\sigma (\theta _ym)\rangle =\int _{{\mathbb {R}}^N}| A^{\frac{\sigma +1}{2}}(\theta _ym)|^2+2\int _{{\mathbb {R}}^N}\nabla m\nabla \theta _y A^\sigma (\theta _ym)\nonumber \\{} & {} \qquad +\langle \Delta \theta _y m, A^\sigma (\theta _y m)\rangle . \end{aligned}$$

(4.60)

Noting that $\sigma <\frac{1}{2},$ by Remark 4.7, we have

$$\begin{aligned}{} & {} 2\int _{{\mathbb {R}}^N}\nabla w\nabla \theta _y A^\sigma (\theta _yw_t)+\langle \Delta \theta _y w, A^\sigma (\theta _y w_t) \rangle \nonumber \\{} & {} \quad \le C_{c_1} \Vert w\Vert _{H_{\textrm{lu}}^1}\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _yw_t)|^2\right) ^{\frac{1}{2}} \nonumber \\{} & {} \quad \le C_{c_1} \Vert w\Vert _{H_{\textrm{lu}}^1}\Vert w_t\Vert _{H_{\textrm{lu}}^1} \nonumber \\{} & {} \quad \le C_{c_1,\varrho _4},\nonumber \\ \end{aligned}$$

(4.61)

$$\begin{aligned}{} & {} 2\delta \int _{{\mathbb {R}}^N}\nabla w\nabla \theta _y A^\sigma (\theta _yw)+\delta \langle \Delta \theta _y w, A^\sigma (\theta _y w)\rangle \nonumber \\{} & {} \quad \le C_{c_1} \Vert w\Vert _{H_{\textrm{lu}}^1}\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _yw)|^2\right) ^{\frac{1}{2}} \nonumber \\{} & {} \quad \le C_{c_1} \Vert w\Vert ^2_{H_{\textrm{lu}}^1} \nonumber \\{} & {} \quad \le C_{c_1,\varrho _4},\nonumber \\ \end{aligned}$$

(4.62)

$$\begin{aligned}{} & {} 2\int _{{\mathbb {R}}^N}\nabla m_{t}\nabla \theta _y A^\sigma (\theta _ym)+\langle \Delta \theta _y m_t, A^\sigma (\theta _y m_t)\rangle \nonumber \\{} & {} \quad \le C_{c_1} \Vert m_{t}\Vert _{H_{\textrm{lu}}^1}\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _y m)|^2\right) ^{\frac{1}{2}} \nonumber \\{} & {} \quad \le C_{c_1} \Vert m_{t}\Vert _{H_{\textrm{lu}}^1}\Vert m\Vert _{H_{\textrm{lu}}^1} \nonumber \\{} & {} \quad \le \varsigma \Vert m_{t}\Vert ^2_{H_{\textrm{lu}}^1} +C_{c_1,\tau }\Vert m\Vert ^2_{H_{\textrm{lu}}^1} \nonumber \\{} & {} \quad \le C_{c_1,\varrho _4,\epsilon }, \end{aligned}$$

(4.63)

and

$$\begin{aligned}{} & {} 2\int _{{\mathbb {R}}^N}\nabla m\nabla \theta _y A^\sigma (\theta _ym)+\langle \Delta \theta _y m, A^\sigma (\theta _y m)\rangle \nonumber \\{} & {} \quad \le C_{c_1} \Vert m\Vert _{H_{\textrm{lu}}^1}\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _y m)|^2\right) ^{\frac{1}{2}} \nonumber \\{} & {} \quad \le C_{c_1} \Vert m\Vert ^2_{H_{\textrm{lu}}^1} \nonumber \\{} & {} \quad \le C_{c_1,\varrho _4,\epsilon }. \end{aligned}$$

(4.64)

Note that $\sigma \le \frac{N+2-(N-2)\gamma }{2},$ by (4.35), we get

$$\begin{aligned}{} & {} |\langle f_1(z),\theta _yA^\sigma (\theta _ym)\rangle |\nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N} \theta _y(1+| z|^\gamma )| A^\sigma (\theta _ym)| \nonumber \\{} & {} \quad \le C\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _ym)|^{\frac{2N}{N-2+2\sigma }}\right) ^{\frac{N-2+2\sigma }{2N}} \left( \int _{B(y,1)}|1+| z|^\gamma |^{\frac{2N}{N+2-2\sigma }}\right) ^{\frac{N+2-2\sigma }{2N}} \nonumber \\{} & {} \quad \le C(1+\Vert z\Vert ^\gamma _{H^1_U})(\Vert \theta _ym\Vert _{L^2}+\Vert A^{\frac{\sigma +1}{2}}(\theta _ym)\Vert _{L^2}), \end{aligned}$$

(4.65)

where $B(y,1)=\{x\in {\mathbb {R}}^N:| x-y|\le 1\}.$

By virtue of (1.5), we have

$$\begin{aligned}{} & {} |\langle f(u)-f(z),\theta _yA^\sigma (\theta _y m)\rangle |\nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N} \theta _y| w|\left( 1+| u|^{\frac{4}{N-2}}+| z|^{\frac{4}{N-2}}\right) | A^\sigma (\theta _ym)| \nonumber \\{} & {} \quad \le C\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _ym)|^{\frac{2N}{N-2+2\sigma }}\right) ^{\frac{N-2+2\sigma }{2N}} \left( \int _{{\mathbb {R}}^N}| |\theta _yw|^{\frac{2N}{N-2-2\sigma }}\right) ^{\frac{N-2-2\sigma }{2N}} \nonumber \\{} & {} \qquad \times \left( \int _{B(y,1)}|1+| u|^{\frac{4}{N-2}}+| z|^{\frac{4}{N-2}}|^{\frac{N}{2}}\right) ^{\frac{2}{N}} \nonumber \\{} & {} \quad \le C\left( \Vert \theta _ym\Vert _{L^2}+\Vert A^{\frac{\sigma +1}{2}}(\theta _y m)\Vert _{L^2}\right) \left( \Vert \theta _yw\Vert _{L^2}+\Vert A^{\frac{\sigma +1}{2}}(\theta _yw)\Vert _{L^2}\right) ,\nonumber \\ \end{aligned}$$

(4.66)

where $\sigma \le \frac{1}{4}< \frac{N-2}{2}$ and $B(y,1)=\{x\in {\mathbb {R}}^N:| x-y|\le 1\}.$

Since $\sigma <\frac{1}{2},$

$$\begin{aligned} \langle g,\theta _yA^\sigma (\theta _ym)\rangle \le \Vert g\Vert _{L_U^2}\Vert m \Vert _{H_U^1}. \end{aligned}$$

(4.67)

Therefore, by virtue of Lemmas 4.1, 4.3 and Remark 4.7, we have

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\left( \int _{{\mathbb {R}}^N} | A^{\frac{\sigma }{2}}(\theta _ym)|^2{+}(1{-}\delta {+}\delta ^2)\int _{{\mathbb {R}}^N} | A^{\frac{\sigma {+}1}{2}}(\theta _yw)|^2{+}\int _{{\mathbb {R}}^N} | A^{\frac{\sigma {+}1}{2}}(\theta _ym)|^2\right) \nonumber \\{} & {} \quad {\le } C\left( \int _{{\mathbb {R}}^N} | A^{\frac{\sigma }{2}}(\theta _ym)|^2{+}(1{-}\delta {+}\delta ^2)\int _{{\mathbb {R}}^N} | A^{\frac{\sigma {+}1}{2}}(\theta _yw)|^2 {+}\int _{{\mathbb {R}}^N} | A^{\frac{\sigma {+}1}{2}}(\theta _ym)|^2\right) \nonumber \\{} & {} \qquad +C_{\Vert g\Vert _{L_U^2 },\varrho _1,\varrho _2,\varrho _3,\varrho _4}. \end{aligned}$$

(4.68)

Applying the Gronwall lemma, we infer that

$$\begin{aligned} \int _{{\mathbb {R}}^N} | A^{\frac{\sigma }{2}}(\theta _ym)|^2+ & {} \int _{{\mathbb {R}}^N} | A^{\frac{\sigma +1}{2}}(\theta _yw)|^2 +\int _{{\mathbb {R}}^N} | A^{\frac{\sigma +1}{2}}(\theta _ym)|^2\nonumber \\{} & {} \le {\mathcal {Q}}_2(\Vert (u_0,u_1)\Vert _{H^1_{\textrm{lu}},\Vert g\Vert _{L_{\textrm{lu}}^2}}){\textrm{e}}^{k t}. \end{aligned}$$

(4.69)

This completes the proof. $\square$

Now, we state our main results:

Theorem 4.9

(Existence of global attractor) Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N).$ Then the semigroup $\{S(t)\}_{t\ge 0}$ generated by the weak solutions of Eqs. (1.1) and (1.2) with the initial data $(u_0,u_1)\in H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$ has an unique $(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N) )$ global attractor ${\mathcal {A}},$ which satisfies : 

• (i) ${\mathcal {A}}$ is closed and compact in $H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N) );$

• (ii) ${\mathcal {A}}$ attracts every bounded subset of $(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$ with respect to $H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N) )$-norms; 

• (iii) ${\mathcal {A}}$ is invariant;  that is,  $S(t){\mathcal {A}}={\mathcal {A}}$ for any $t\ge 0.$

5 Asymptotic Regularity

In this section, we will prove the regularity of the $(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$$\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N) )$ global attractor by some bootstrap arguments. Similar to that in Zelik [30, 31], based on Lemmas 4.6 and 4.8 above, for the solution $(u(t),u_t(t)),$ we can decompose it as follows.

Lemma 5.1

Assume that f satisfies (1.3)–(1.5) and $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N),$ and let u(t) be the solution of Eqs. (1.1)–(1.2) with the initial data $(u_0,u_1)\in B.$ Then for any $\varsigma >0,$ there are positive constants $C_{\varsigma }$ and $K_{\varsigma },$ such that

$$\begin{aligned} u(t)=z_1(t)+w_1(t),\quad \text {for all }t\ge 0, \end{aligned}$$

(5.1)

where $z_1(t),$ $w_1(t)$ satisfy the estimates as follows : 

$$\begin{aligned} \Vert w_1(t)\Vert ^2_{H_U^{1+\sigma }}\le K_{\varsigma },\quad t\ge 0, \end{aligned}$$

(5.2)

and for every $t\ge s\ge 0,$

$$\begin{aligned} \int _s^t\Vert z_1(r)\Vert _{H_U^{1}}^2 \textrm{d}r\le \varsigma (t-s)+C_{\varsigma }, \end{aligned}$$

(5.3)

where the constants $C_{\varsigma }$ and $K_{\varsigma }$ depend on $\varsigma ,$ $\sigma .$

Proof

Note that

$$\begin{aligned} {\mathcal {B}}=\bigcup _{t\ge T_{B_0}}S(t)B_0; \end{aligned}$$

by Lemma 4.1, we infer that

$$\begin{aligned} \sup _{t\ge 0}\Vert S(t)(u_0,u_1)\Vert ^2_{H^{1}_{\textrm{lu}}}\le \varrho _1, \quad \text {for all }(u_0,u_1) \in {\mathcal {B}}. \end{aligned}$$

Now, taking $T\ge \frac{1}{k_0}\ln \frac{{\mathcal {Q}}_1(\varrho _1)}{\varepsilon }$(where ${\mathcal {Q}}_1(\cdot )$ the function in Lemma 4.6), and in every interval $[mT,(m+1)T),~m=1,2,\ldots ,$ we set

$$\begin{aligned} z_1(t)=z(t)\quad \text {and}\quad w_1(t)=w(t), \end{aligned}$$

where z(t) is the solution of Eq. (4.38) in the interval $[(m-1)T,(m+1)T)$ with the initial data $(z((m-1)T),z_t((m-1)T))=(u((m-1)T),u_t((m-1)T)),$ and w(t) is the solution of Eq. (4.39) in the interval $[(m-1)T,(m+1)T)$ with the initial data $(w((m-1)T),w_t((m-1)T))=(0,0).$

In the interval [0, T),  we set

$$\begin{aligned} z_1(t)=z(t)\quad \text {and}\quad w_1(t)=w(t), \end{aligned}$$

where z(t) is the solution of Eq. (4.38) with the initial data $(z(0),z_t(0))=(u_0,u_1),$ and $(w(t),w_t(t))$ is the solution of Eq. (4.39) with the initial data $(w(0),w_t(0))=(0,0).$

Then from Lemma 4.6, we infer that

$$\begin{aligned} \int _s^t\Vert z_1(r)\Vert ^2_{H_U^1} {\textrm{d}}r\le \varsigma (t-s)+\chi _{[0,T)}(s){\mathcal {Q}}_1(\varrho _1),\quad \text {for all }t\ge s\ge 0, \end{aligned}$$

where $\chi _{[0,T)}(s)$ is the characteristic function of set [0, T). According to Lemma 4.8, we infer that

$$\begin{aligned} \Vert w_1(t)\Vert ^2_{H_U^{1+\sigma }} \le {\mathcal {Q}}_2(\Vert (u_0,u_1)\Vert _{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1},\Vert g\Vert _{L^2_{\textrm{lu}}}){\textrm{e}}^{2k_1 T},\quad \text {for all }t\ge 0. \end{aligned}$$

This completes the proof. $\square$

Remark 5.2

According to the proof of Lemma 5.1, we infer that the decomposition $z_1(t)$ can also further satisfy that

$$\begin{aligned} \Vert z_1(t)\Vert ^2_{H_U^1}\le {\mathcal {Q}}_1(\varrho _1),\quad \text {for all } t\ge 0. \end{aligned}$$

Lemma 5.3

Assume that f satisfies (1.3)–(1.5), and $f_1$ satisfies (4.35)–(4.37), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N).$ For any bounded set $B\subset H_{\textrm{lu}}^1({\mathbb {R}}^N)\times H_{\textrm{lu}}^1({\mathbb {R}}^N),$ there exists a positive constant $J_{\Vert B\Vert _{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1}}$ which depends only on the $H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1$-bounds of B,  such that

$$\begin{aligned} \Vert K(t)(u_0,u_1)\Vert _{H_{\textrm{lu}}^{1+\sigma }}^2\le J_{\Vert B\Vert _{H_{\textrm{lu}}^1\times H_{\textrm{lu}}^1}},\quad \text {for all }t\ge 0\ \text {and}\ (u_0,u_1)\in B, \end{aligned}$$

(5.4)

where $\sigma$ is given in Lemma 4.8.

Proof

Multiplying (4.54) by $\theta _yA^\sigma (\theta _ym),$ similar to the proof of Lemma 4.8, here we only need to deal with the nonlinear term in a different way:

$$\begin{aligned}{} & {} \langle f(u)-f(z)+f_1(z),\theta _yA^\sigma (\theta _ym)\rangle \nonumber \\{} & {} \quad =\langle f(u)-f(z)+f_1(z),\theta _yA^\sigma (\delta \theta _yw+\theta _yw_t)\rangle . \end{aligned}$$

(5.5)

By (1.5), we have

$$\begin{aligned}{} & {} \langle f(u)-f(z)+f_1(z),\delta \theta _yA^\sigma (\theta _yw)\rangle \nonumber \\{} & {} \quad =\langle f(u)-f(z),\delta \theta _yA^\sigma (\theta _yw)\rangle +\langle f_1(z),\delta \theta _yA^\sigma (\theta _yw)\rangle \nonumber \\{} & {} \quad \le C\int _{{\mathbb {R}}^N}(1+| u|^{\frac{4}{N-2}}+| z|^{\frac{4}{N-2}})\theta _y| w|| A^\sigma (\theta _yw)|+\langle f_1(z),\theta _yA^\sigma (\theta _yw)\rangle \nonumber \\{} & {} \quad \triangleq \sum _{i=1}^4I_i. \end{aligned}$$

(5.6)

For $I_1,$ we infer that

$$\begin{aligned} I_1=C\int _{{\mathbb {R}}^N}\theta _y| w|| A^\sigma (\theta _yw)|\le \Vert w\Vert _{L_U^2}\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _y w)|^2\right) ^{\frac{1}{2}}. \end{aligned}$$

(5.7)

For $I_2,$ using Lemma 4.1, we get

$$\begin{aligned} I_2= & {} C\int _{{\mathbb {R}}^N}| u|^{\frac{4}{N-2}}\theta _y| w|| A^\sigma (\theta _yw)|\nonumber \\\le & {} C\int _{{\mathbb {R}}^N}(| z_1|^{\frac{4}{N-2}}+| w_1|^{\frac{4}{N-2}})\theta _y| w|| A^\sigma (\theta _yw)|. \end{aligned}$$

(5.8)

By Remark 4.7 and the interpolation inequality, we have

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^N}| z_1|^{\frac{4}{N-2}}\theta _y| w|| A^\sigma (\theta _yw)|\nonumber \\{} & {} \quad =\int _{B(y,1)}| z_1|^{\frac{4}{N-2}}| \theta _yw|| A^\sigma (\theta _yw)|\nonumber \\{} & {} \quad \le C\left( \int _{B(y,1)}| z_1|^{\frac{2N}{N-2}}\right) ^{\frac{2}{N}}\left( \int _{{\mathbb {R}}^N}| \theta _yw|^{\frac{2N}{N-2-2\sigma }}\right) ^{\frac{N-2-2\sigma }{2N}}\nonumber \\{} & {} \qquad \times \left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _yw)|^{\frac{2N}{N-2+2\sigma }}\right) ^{\frac{N-2+2\sigma }{2N}} \nonumber \\{} & {} \quad {\le } C\Vert z_1\Vert _{H_U^1}^{\frac{4}{N{-}2}}(\Vert A^{\frac{1{+}\sigma }{2}}(\theta _yw)\Vert _{L^2}{+}\Vert \theta _yw\Vert _{L^2})(\Vert A^{\frac{1{+}\sigma }{2}}(\theta _yw)\Vert _{L^2}{+}\Vert A^\sigma (\theta _yw)\Vert _{L^2}) \nonumber \\{} & {} \quad {\le } C_{\varrho _1,\epsilon }\Vert z_1\Vert _{H_U^1}^2\int _{{\mathbb {R}}^N}| A^{\frac{1{+}\sigma }{2}}(\theta _yw)|^2{+}\epsilon \int _{{\mathbb {R}}^N}| A^{\frac{1{+}\sigma }{2}}(\theta _yw)|^2 {+}C_{\varrho _1,\epsilon }(1{+}\Vert \theta _yw\Vert _{L^2}^2),\qquad \quad \end{aligned}$$

(5.9)

and by Lemma 5.1 and the interpolation inequality, we infer that

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^N}| w_1|^{\frac{4}{N-2}}\theta _y| w|| A^\sigma (\theta _yw)|\le C\left( \int _{B(y,1)}| w_1|^{\frac{2N}{N-2-2\sigma }}\right) ^{\frac{2(N-2-2\sigma )}{N(N-2)}}\nonumber \\{} & {} \qquad \times \left( \int _{{\mathbb {R}}^N}| \theta _yw|^{\frac{2N(N-2)}{(N-2)^2-2(N-6)\sigma }}\right) ^{\frac{(N-2)^2-2(N-6) \sigma }{2N(N-2)}}\left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _yw)|^{\frac{2N}{N-2+2\sigma }}\right) ^{\frac{N-2+2\sigma }{2N}} \nonumber \\{} & {} \quad \le C\Vert w_1\Vert _{H_U^{1+\sigma }}^{\frac{4}{N-2}}\Vert \theta _y w\Vert _{L^{\frac{2N(N-2)}{(N-2)^2-2(N-6)\sigma }}}(\Vert A^{\frac{1+\sigma }{2}}(\theta _yw)\Vert _{L^2}+\Vert A^\sigma (\theta _yw)\Vert _{L^2}) \nonumber \\{} & {} \quad \le \epsilon \int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2+CK_{\varsigma ,\epsilon }(1+\Vert \theta _yw\Vert _{L^2}^2). \end{aligned}$$

(5.10)

Hence,

$$\begin{aligned}{} & {} I_2\le \epsilon \int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2+C_{\varrho _1,\epsilon }\Vert z_1\Vert _{H_U^1}^2\int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2\nonumber \\{} & {} \quad +C_{K_{\varsigma },\varrho _1,\epsilon }(1+\Vert \theta _yw\Vert _{L^2}^2). \end{aligned}$$

(5.11)

For $I_3,$ we get

$$\begin{aligned} I_3= & {} C\int _{{\mathbb {R}}^N}| z|^{\frac{4}{N-2}}\theta _y| w|| A^\sigma (\theta _yw)| C\left( \int _{B(y,1)}| z|^{\frac{2N}{N-2}}\right) ^{\frac{2}{N}}\nonumber \\{} & {} \qquad \times \left( \int _{{\mathbb {R}}^N}| \theta _yw|^{\frac{2N}{N-2-2\sigma }}\right) ^{\frac{N-2-2\sigma }{2N}} \left( \int _{{\mathbb {R}}^N}| A^\sigma (\theta _yw)|^{\frac{2N}{N-2+2\sigma }}\right) ^{\frac{N-2+2\sigma }{2N}} \nonumber \\\le & {} C\Vert z\Vert _{H_U^1}^{\frac{4}{N-2}}\left( \left\| A^{\frac{1+\sigma }{2}}(\theta _yw)\right\| _{L^2}+\Vert \theta _yw\Vert _{L^2}\right) \nonumber \\{} & {} \qquad \times \left( \Vert A^{\frac{1+\sigma }{2}}(\theta _yw)\Vert _{L^2}+\Vert A^\sigma (\theta _yw)\Vert _{L^2}\right) \nonumber \\\le & {} C_{\varrho _1}\Vert z\Vert _{H_U^1}^{\frac{4}{N-2}}\int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2 +C_{\varrho _1}(1+\Vert \theta _yw\Vert _{L^2}^2). \end{aligned}$$

(5.12)

Note that from Lemma 5.1 and Remark 5.2, we can take T large enough such that

$$\begin{aligned} \Vert z\Vert _{H_U^1}^{\frac{4}{N-2}}\le \frac{\epsilon }{C_{\varrho _1}},\quad \text {for all } t\ge T. \end{aligned}$$

(5.13)

For $I_4,$ by (4.35), we infer that

$$\begin{aligned} I_4= & {} \langle f_1(z),\theta _yA^\sigma (\theta _yw)\rangle \nonumber \\\le & {} C\int _{{\mathbb {R}}^N}(1+| z|^\gamma )\theta _y| A^\sigma (\theta _yw)| \nonumber \\\le & {} C\left( 1+\int _{B(y,1)}\theta _y| z|^{\frac{2N\gamma }{N+2-2\sigma }}\right) ^{\frac{N+2-2\sigma }{2N}}\Vert A^\sigma (\theta _yw)\Vert _{L^{\frac{2N}{N-2+2\sigma }}} \nonumber \\\le & {} C(1+\Vert z\Vert ^\gamma _{H_U^1})\left( \Vert A^{\frac{1+\sigma }{2}}(\theta _yw)\Vert _{L^2}+\Vert A^\sigma (\theta _yw)\Vert _{L^2}\right) \nonumber \\\le & {} C_{\varrho _3,\epsilon }(1+\Vert \theta _yw\Vert ^2_{L^2})+\epsilon \int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2. \end{aligned}$$

(5.14)

Thus,

$$\begin{aligned}{} & {} \langle f(u)-f(z)+f_1(z),\delta \theta _yA^\sigma (\theta _yw)\rangle \nonumber \\{} & {} \quad \le 5\epsilon \int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2+ C_{\varrho _1,\epsilon }\Vert z_1\Vert _{H_U^1}^2\int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2 \nonumber \\{} & {} \qquad +C_{K_{\varsigma },\varrho _3,\epsilon }(1+\Vert \theta _yw\Vert _{L^2}^2). \end{aligned}$$

(5.15)

Similarly, we get that

$$\begin{aligned}{} & {} \langle f(u)-f(z)+f_1(z),\theta _yA^\sigma (\theta _yw_t)\rangle \nonumber \\{} & {} \quad \le 5\epsilon \left( \int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2+\int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw_t)|^2\right) \nonumber \\{} & {} \qquad +C_{\varrho _3,\epsilon }\Vert z_1\Vert _{H_U^1}^2\int _{{\mathbb {R}}^N}| A^{\frac{1+\sigma }{2}}(\theta _yw)|^2 +C_{K_{\varsigma },\varrho _3,\epsilon }(1+\Vert \theta _yw\Vert _{L^2}^2+\Vert \theta _yw_t\Vert _{L^2}^2).\nonumber \\ \end{aligned}$$

(5.16)

We denote that

$$\begin{aligned} {\mathcal {E}}_2(t){=}\int _{{\mathbb {R}}^N} | A^{\frac{\sigma }{2}}(\theta _ym)|^2{+}(1{-}\delta {+}\delta ^2)\int _{{\mathbb {R}}^N} | A^{\frac{\sigma +1}{2}}(\theta _yw)|^2+\int _{{\mathbb {R}}^N} | A^{\frac{\sigma +1}{2}}(\theta _ym)|^2). \end{aligned}$$

Therefore, we get that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}{\mathcal {E}}_2(t) +C_{\epsilon }(1-C_{\varrho _1}\Vert z_1\Vert _{H_U^1}^2){\mathcal {E}}_2(t) \le C_{K_{\varsigma },\Vert g\Vert _{L_U^2 },\varrho _1,\varrho _2,\varrho _3,\varrho _4}. \end{aligned}$$

(5.17)

Applying the Gronwall Lemma and integrating over $[1+T,t],$ we infer

$$\begin{aligned} {\mathcal {E}}_2(t)\le & {} {\textrm{e}}^{-\int ^t_{T+1}C_{\epsilon }(1-C_{\varrho _1}\Vert z_1(s)\Vert _{H_U^1}^2){\textrm{d}}s}{\mathcal {E}}_2(T+1)\nonumber \\{} & {} +C_{K_{\varsigma },\Vert g\Vert _{L_U^2 },\varrho _1,\varrho _2,\varrho _3,\varrho _4}\int ^t_{T+1}{\textrm{e}}^{\int _t^sC_{\epsilon }(1-C_{\varrho _1}\Vert z_1(\tau )\Vert _{H_U^1}^2)\textrm{d}\tau }\textrm{d}s.\nonumber \\ \end{aligned}$$

(5.18)

According to Lemma 5.1, for every $t\ge s\ge 0,$

$$\begin{aligned} \int _s^t\Vert z_1(r)\Vert _{H_U^{1}}^2 \textrm{d}r\le \varsigma (t-s)+C_{\varsigma }. \end{aligned}$$

Now, we choose $\varsigma <\frac{1}{2 C_{\varrho _1}}$ and have

$$\begin{aligned}{} & {} C_{K_{\varsigma },\Vert g\Vert _{L_U^2 },\varrho _1,\varrho _2,\varrho _3,\varrho _4}\int ^t_{T+1}{\textrm{e}}^{\int _t^sC_{\epsilon }(1-C_{\varrho _1}\Vert z_1(\tau )\Vert _{H_U^1}^2)d \tau }\textrm{d}s \nonumber \\{} & {} \quad \le C_{K_{\varsigma },\Vert g\Vert _{L_U^2 },\varrho _1,\varrho _2,\varrho _3,\varrho _4} {\textrm{e}}^{C_{\varrho _1}C_{\epsilon }}\int ^t_{T+1}{\textrm{e}}^{C_{\epsilon }(1-C_{\varrho _1}\varsigma )(s-t)}\textrm{d}s \nonumber \\{} & {} \quad \le C_{K_{\varsigma },\Vert g\Vert _{L_U^2 },\epsilon ,\varrho _1,\varrho _2,\varrho _3,\varrho _4}\int ^t_{T+1}{\textrm{e}}^{\frac{s-t}{2}}\textrm{d}s\nonumber \\{} & {} \quad \le C_{K_{\varsigma },\Vert g\Vert _{L_U^2 },\epsilon ,\varrho _1,\varrho _2,\varrho _3,\varrho _4}, \end{aligned}$$

(5.19)

and

$$\begin{aligned}{} & {} {\textrm{e}}^{-\int ^t_{T+1}C_{\epsilon }(1-C_{\varrho _1}\Vert z_1(s)\Vert _{H_U^1}^2)\textrm{d}s}{\mathcal {E}}_2(T+1)\nonumber \\{} & {} \quad \le {\textrm{e}}^{-\frac{1}{2}(t-T-1)}{\textrm{e}}^{C_{\varrho _1,\varsigma ,\epsilon }}{\mathcal {E}}_2(T+1). \end{aligned}$$

(5.20)

Note that T is fixed, and using Lemma 4.8, we complete the proof. $\square$

Lemma 5.4

Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N).$ Assume $B_{\sigma }$ is an arbitrary bounded set in $H_{\textrm{lu}}^{1+\sigma }\times H_{\textrm{lu}}^{1+\sigma }.$ Then there exists a constant $M_{\sigma }(>0)$ which depends only on the $H_{\textrm{lu}}^{1+\sigma }\times H_{\textrm{lu}}^{1+\sigma }$-bound of $B_{\sigma }$ such that

$$\begin{aligned} \Vert S(t)(u_0,u_1)\Vert _{H_{\textrm{lu}}^{1+\sigma }\times H_{\textrm{lu}}^{1+\sigma }}^2\le M_{\sigma },\quad \text {for all }t\ge 0\ \text {and}\ (u_0,u_1)\in B_{\sigma }. \end{aligned}$$

(5.21)

Proof

The proof of this lemma is completely similar to that of Lemma 5.3, and we can deal with the nonlinear term by similar calculations used in Lemma 5.3, so we omit it here. $\square$

In the following, we can perform the bootstrap arguments to obtain the asymptotic regularity of the solutions. Similar to the proof Lemma 5.3, we infer the following two lemmas.

Lemma 5.5

Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N)$ and $\sigma \le \theta \le 1.$ Then for any bounded $B_{\theta }\subset H_{\textrm{lu}}^{1+\theta }\times H_{\textrm{lu}}^{1+\theta }.$ Then there exists a constant $M_{\theta }(>0)$ which depends only on the $H_{\textrm{lu}}^{1+\theta }\times H_{\textrm{lu}}^{1+\theta }$-bound of $B_{\theta }$ such that

$$\begin{aligned}{} & {} \Vert S(t)(u_0,u_1)\Vert _{H_{\textrm{lu}}^{1+\theta }\times H_{\textrm{lu}}^{1+\theta }}^2\le M_{\theta },\nonumber \\{} & {} \quad \text {for all }t\ge 0\ \text {and}\ (u_0,u_1)\in B_{\theta }. \end{aligned}$$

(5.22)

Lemma 5.6

Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N)$ and $\theta \in [\sigma ,1-\min \{\sigma ,\frac{4\sigma }{n-2}\}],$ and assume that the initial data set $B_{\theta }$ is bounded in $H_{\textrm{lu}}^{1+\theta }\times H_{\textrm{lu}}^{1+\theta },$ then the decomposed ingredient $(w(t),w_t(t))$ satisfies that

$$\begin{aligned} \Vert K(t)(u_0,u_1)\Vert _{H^{1+\theta +s_0}_{\textrm{lu}}}^2\le J_{\theta },\quad \text {for all }t\ge 0\ \text {and}\ (u_0,u_1)\in B_{\theta }, \end{aligned}$$

(5.23)

where $s_0=\min \{\sigma ,\frac{4\sigma }{n-2}\}$ and the constant $J_{\theta }(>0)$ which depends only on the $H_{\textrm{lu}}^{1+\theta }\times H_{\textrm{lu}}^{1+\theta }$-bound of $B_{\theta }.$

We also need the following attraction transitivity lemma.

Lemma 5.7

[17] Let $K_1,$ $K_2,$ $K_3$ be subsets of H such that

$$\begin{aligned} {\textrm{dist}}_H(S(t)K_1,K_2)\le L_1 {\textrm{e}}^{-\nu _1 t},~{\textrm{dist}}_H(S(t)K_2,K_3)\le L_2 {\textrm{e}}^{-\nu _2 t}, \end{aligned}$$

for some $\nu _1,~\nu _2>0$ and $L_1,~L_2>0.$ Assume that for all $z_1,z_2\in \bigcup _{t\ge 0}S(t)K_j$$(j=1,2,3),$ there holds

$$\begin{aligned} \Vert S(t)z_1-S(t)z_2\Vert \le L_0 {\textrm{e}}^{\nu _0 t}\Vert z_1-z_2\Vert \end{aligned}$$

for some $\nu _0>0$ and some $L_0>0.$ Then it follows that

$$\begin{aligned} {\textrm{dist}}_H(S(t)K_1,K_3)\le L {\textrm{e}}^{-\nu t}, \end{aligned}$$

where $\nu =\frac{\nu _1\nu _2}{\nu _0+\nu _1+\nu _2}$ and $L=L_0L_1+L_2.$

Now, we state the following asymptotic regularity results:

Theorem 5.8

(Asymptotic Regularity) Assume that f satisfies (1.3)–(1.5), $g(x)\in L_{\textrm{lu}}^2({\mathbb {R}}^N),$ and let $\{S(t)\}_{t\ge 0}$ be the semigroup generated by the weak solutions of Eqs. (1.1)–(1.2) with the initial data $(u_0,u_1)\in H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N).$ Then,  there exists a set ${\mathcal {B}}\subset H_{\textrm{lu}}^2({\mathbb {R}}^N)\times H_{\textrm{lu}}^2({\mathbb {R}}^N)$ (closed and bounded in $H_{\textrm{lu}}^2({\mathbb {R}}^N)\times H_{\textrm{lu}}^2({\mathbb {R}}^N)),$ a positive constant $\nu$ and a monotonically increasing function ${\mathcal {Q}}(\cdot )$ such that :  for any bounded set $B\subset H_{\textrm{lu}}^1({\mathbb {R}}^N)\times H_{\textrm{lu}}^1({\mathbb {R}}^N),$ the following estimate holds : 

$$\begin{aligned} {\textrm{dist}}_{*}(S(t)B,{\mathcal {B}})\le {\mathcal {Q}}(\Vert B\Vert _{H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)}){\textrm{e}}^{-\nu t}, \end{aligned}$$

where ${\textrm{dist}}_*$ denotes the usual Hausdorff semidistance in $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N).$

Proof

We denote

$$\begin{aligned} {\mathcal {B}}=\bigcup _{t\ge T_{B_0}}S(t)B_0, \end{aligned}$$

where $B_0$ be the bounded absorbing set stated in Remark 4.2 and $T_{B_0}=\max \{T_1(B),T_2(B),T_3(B)\}.$

According to Lemmas 4.6 and 5.3, we know that there is a set $A_{\sigma }$ which is bounded in $H^{1+\sigma }_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1+\sigma }_{\textrm{lu}}({\mathbb {R}}^N)$ such that

$$\begin{aligned} {\textrm{dist}}_{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}(S(t){\mathcal {B}},A_{\sigma })\le {\textrm{dist}}_{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}(D(t){\mathcal {B}},A_{\sigma })\le {\mathcal {Q}}_1(\Vert {\mathcal {B}}\Vert _{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}){\textrm{e}}^{-k_0 t}. \end{aligned}$$

Applying Lemmas 4.6 and 5.6 to $A_{\sigma },$ we see that there is a set $A_{\sigma +s}$ which is bounded in $H^{1+\sigma +s}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1+\sigma +s}_{\textrm{lu}}({\mathbb {R}}^N),$ such that

$$\begin{aligned} {\textrm{dist}}_{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}(S(t){\mathcal {B}},A_{\sigma +s})\le {\textrm{dist}}_{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}(D(t){\mathcal {B}},A_{\sigma +s})\le {\mathcal {Q}}_1(\Vert {\mathcal {B}}\Vert _{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}){\textrm{e}}^{-k_0 t}, \end{aligned}$$

where $k_0$ depends only on the $H_{\textrm{lu}}^{1}\times H_{\textrm{lu}}^{1}$-bound of $A_{\sigma }.$ Combining this with Remark 3.2, we know that the conditions in Lemma 5.7 are all satisfied. Hence we have

$$\begin{aligned} {\textrm{dist}}_{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}(S(t){\mathcal {B}},A_{\sigma +s})\le C {\mathcal {Q}}_1(\Vert {\mathcal {B}}\Vert _{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}){\textrm{e}}^{-k_0 t}, \end{aligned}$$

(5.24)

for two appropriate constants C and $k_0.$

Note that $\sigma =\min \{\frac{1}{4},\frac{N+2-(N-2)\gamma }{2}\}$ and $s_0=\min \{\sigma ,\frac{4\sigma }{n-2}\}$ are fixed, by finite steps (e.g., at most by $[\frac{1}{s}]+2$ steps) we can infer that there is a bounded (in $H^{2}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{2}_{\textrm{lu}}({\mathbb {R}}^N))$ set $B_1\subset H^{2}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{2}_{\textrm{lu}}({\mathbb {R}}^N)$ such that

$$\begin{aligned} {\textrm{dist}}_{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}(S(t){\mathcal {B}},B_1)\le {\mathcal {Q}}_1(\Vert {\mathcal {B}}\Vert _{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}){\textrm{e}}^{-\nu t}. \end{aligned}$$

Now, for any bounded set $B\in H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),$ by Lemma 4.1 and Remark 4.2, we see that there exist a T such that

$$\begin{aligned} S(t)B\subset {\mathcal {B}},\quad \text {for all }t\ge T. \end{aligned}$$

Hence,

$$\begin{aligned} {\textrm{dist}}_{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}}(S(t){\mathcal {B}},B_1)\le Me^{\nu T}{\textrm{e}}^{-\nu t}, \end{aligned}$$

(5.25)

where $M=\sup \{\Vert S(t)B\Vert _{H^{1}_{\textrm{lu}}\times H^{1}_{\textrm{lu}}},~0\le t\le T\}<\infty .$

Now, we apply the attraction transitivity lemma, i.e., Lemma 5.7, then again to (5.24) and (5.25), and this completes the proof. $\square$

Remark 5.9

The $(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N))$-global attractor given in Theorem 4.9 is bounded in the locally uniform space $H^{2}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{2}_{\textrm{lu}}({\mathbb {R}}^N),$ which appears to be optimal.

Remark 5.10

There exists a bounded (in $(H^{2}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{2}_{\textrm{lu}}({\mathbb {R}}^N)$) subset which attracts exponentially every initial $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$-bounded set with respect to the $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$-norm.

Remark 5.11

To our best knowledge, this is the first time we obtain the regularity for Eqs. (1.1) and (1.2) with critical nonlinearity on the unbounded domain. Maybe it is a basis for further considering the asymptotic behavior, e.g., based on this result, whether the exponential attractors exist for Eqs. (1.1) and (1.2) with critical nonlinearity on unbounded domain is still open.