1 Introduction

In this paper we are concerned with the following weakly damped wave equation with gradient type nonlinearity on a bounded domain \(\Omega \subset {\mathbb {R}}^{n}\) with smooth boundary \({\partial \Omega}\)

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle u_{tt}+2\alpha u_{t}=\triangle u-|u|^{p-1}u+f(u,\nabla u,x,t), &{}in\ \Omega \times {\mathbb {R}}^{+},\\ \displaystyle u(x,t)=0,&{} on\ \partial \Omega \times {\mathbb {R}}^{+},\\ \displaystyle u(x,0)=\varphi (x), \ \ u_{t}(x,0)=\psi (x), &{} in\ \Omega , \end{array} \right. \end{aligned}$$
(1.1)

where \(\alpha>0, p>1,\) and \(1<p<\frac{n}{n-2}, n\ge 3; 1<p<\infty , n=1,2,\) respectively. The source \(f\in C^1\) and satisfies the following growth condition

$$\begin{aligned} |f(u,\nabla u,x,t)|\le C_1|\nabla u|^s+C_2|u|^q+g(x,t), \end{aligned}$$
(1.2)

where \(g\in L^2(\Omega \times (0,T)), q\le \frac{p+1}{2}, s\le 1\) for some constant \(C_1,C_2>0\).

Similar initial-boundary problem was investigated by Meng and Zhu in [18, 28] for the equation

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle u_{tt}+\alpha u_{t}-\triangle u+\varphi (u)=f, &{}in\ \Omega \times {\mathbb {R}}^{+},\\ \displaystyle u(x,t)=0,&{} on\ \partial \Omega \times {\mathbb {R}}^{+},\\ \displaystyle u(\cdot ,0)=u_0, \ \ u_{t}(\cdot ,0)=u_1, &{} in\ \Omega , \end{array} \right. \end{aligned}$$
(1.3)

where \(\alpha >0, \varphi\) is the nonlinear term, and f is a given external forcing term, \(\Omega \subset {\mathbb {R}}^3\) is a bounded domain with smooth boundary \(\partial \Omega\). Nonlinear wave equation of the type (1.3) arises as an evolutionary mathematical model in many branched of physics, for example, (i) modeling a continuous Josephson junction with \(\varphi (u)=\beta \sin u\); (ii) modeling a relativistic quantum mechanics with \(\varphi (u)=|u|^\gamma u\). A relevant problem is to investigate the asymptotic dynamical behavior of these mathematical models. The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics(see,e.g. [14, 21] and references therein). One way to treat this problem is to analyse the existence of its global attractor. Meng obtained the existence of a global attractor for (1.3) in strong topological space \(H^2(\Omega )\cap H_0^1(\Omega )\times H_0^1(\Omega )\) where the nonlinear term \(\varphi\) with some polynomial growth and the external forcing term f is independent of time. Zhu invested the case which \(\varphi (u)=0\) with nonlinear boundary conditions. The boundary was divided into two parts: one part is Dirichlet boundary and the other part is with boundary damping which the framework was arising in control theory in [12, 13]. And also, the other way to treat this problem is to concern the decay of solutions (see,e.g. [4, 19, 23] and references therein).

Attractor is an important concept describing asymptotic properties of dynamical systems, a great deal of work has been devoted to the existence of global attractors of dynamical systems (see, e.g. [1, 3, 5, 6, 8,9,10, 22, 26, 27] and references therein). The existence of a global attractor Eq. (1.1) which the source term only containing f was proved by Hale [8] for f satisfying for \(n\ge 3\) the growth condition \(f(u)\le C_{0}(|u|^{\gamma }+ 1)\), with \(1\le \gamma <\frac{n}{n-2}\). For the case \(n=2\), Hale and Raugel [7] proved the existence of the attractor under an exponential growth condition of the type \(|f(u)|\le \exp \theta (u)\) (such a condition previously appearing in the work of Gallou\(\ddot{e}\)t [16]). The existence of the attractor in the critical case \(\gamma =\frac{n}{n-2}\) was first proved by Babin and Vishik [2], and then more generally by Arrieta, Carvalho and Hale [5]. For other treatments see Chepyzhov and Vishik [5], Ladyzhenskaya [11], Raugel [24] and Temam [26]. When \(\Omega\) is bounded and u is subjected to suitable boundary conditions, the general result is that the dynamical system associated with the problem possesses a global attractor in natural energy space \(H_{0}^{1}(\Omega )\times L^{2}(\Omega )\) if nonlinear term f has subcritical or critical exponent, because there exists typical parabolic-like flows with an inherent smoothing mechanism. However, the nonlinear source f with gradient in Eq. (1.1) which is different from [3, 6, 27, 28]. So we make the nonlinear source f those under some growth conditions controlled by the variational structure. By the traditional method [25] (for examples), in order to obtain the existence of global attractors for semilinear wave equations, one needs to verify the uniform compactness of the semigroup by getting the boundedness in a more regular function space. However, in some cases it is difficult to obtain the uniform compactness of the semigroup. Fortunately, a new method for obtaining the global attractors has been developed in [17]. With this method, one only needs to verify a necessary compactness condition(\(\omega\)-limit compactness) with the same type of energy estimates as those for establishing the absorbing sets. In this paper, we use this method to obtain the existence of global attractors for the Eq. (1.1) with the complicated condition where the source term \(f(u,\nabla u,x,t)\) is without variational structure.

This paper is organized as follows:

-in Sect. 2 we obtained the existence and uniqueness of weak solution by using Galerkin method and compulsively variational method;

-in Sect. 3 we obtained the existence of global attractor for the Eq. (1.1) by using the new method (\(\omega\)-compactness condition).

2 Existence and Uniqueness of the Global Weak Solution

Definition 2.1

(Weak Solution) [20]  A \(u=u(x,t)\) is called a weak solution of the Eq. (1.1), if \(u\in W_{loc}^{1,\infty }(0,T;L^{2}(\Omega ))\cap L_{loc}^{\infty }(0,T;H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega ))\) and satisfies the Eq. (1.1) in the distribution sense, i.e.

$$\begin{aligned} {<u_{t},v>}_{2}+2\alpha<u,v>_{2}=\int _{0}^{t}<Gu,v>d\tau +2\alpha {<\varphi ,v>}_{2}+{<\psi ,v>}_{2} \end{aligned}$$

for any \(v\in H_{0}^{1}(\Omega ), t\in (0,T)\), where \((\varphi ,\psi )\in (H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega ))\times L^{2}(\Omega ),\) \(<\cdot ,\cdot >_2\) means the inner product \(<\cdot ,\cdot >_{L^{2}(\Omega )}\) and \(G(u)=\triangle u-|u|^{p-1}u+f(u,\nabla u,x,t).\)

Theorem 2.1

  (Existence) Under condition (1.2), for any \((\varphi , \psi )\in (H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega ))\times L ^{2}(\Omega )\), then the Eq. (1.1) has a global weak solution

$$\begin{aligned} u\in W_{loc}^{1,\infty }(0,\infty ;L^{2}(\Omega ))\bigcap L_{loc}^{\infty }(0,\infty ;H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega )). \end{aligned}$$

Proof

Fix spaces as follows:

$$\begin{aligned} X_{2}= X_{1}=H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega ), X=C_{0}^{\infty }(\Omega ), \\ H_{1}=H=L^{2}(\Omega ), H_{2}=H^{2}(\Omega ), \end{aligned}$$

\(X_{\frac{1}{2}}\) is a fractional space which generated by \(\Delta ^{\frac{1}{2}}, X_{\frac{1}{2}}=H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega ).\)

Set \(\{e_{k}|k=1,2,\ldots \}\subset X\) is the common orthogonal basis of H and \(H_2\), that is

$$\begin{aligned} \left\{ \ \begin{aligned}&H_{n}=\left\{\sum _{k=1}^{n}\xi _{k}e_{k}|\xi _{k}\in R^{1}, 1\le k\le n\right\},\\&{\widetilde{H}}_{n}=\left\{\sum _{k=1}^{n}\eta _{k}(t)e_{k}|\eta _{k}(t)\in C^{2}[0, \infty ), 1\le k\le n\right\}.\\ \end{aligned}\right. \end{aligned}$$
(2.1)

Note \(u_n=\sum _{k=1}^{n}u_k(t)e_k\). Multiply (1.1) by \(e_k\) and integrate it over \(\Omega\) to get

$$\begin{aligned} \left\{ \ \begin{aligned}&\frac{d^{2}u_{k}(t)}{dt^{2}}+2\alpha \frac{du_{k}(t)}{dt}=<\triangle u_{n}-|u_{n}|^{p-1}u_{n}+f, e_{k}>_{H}, ~~ 0\le k\le n\\&u_{k}(0)=<\varphi _{n}, e_{k}>_{H}, ~~\frac{du_{k}(0)}{dt}=<\psi _{n}, e_{k}>_{H}.\\ \end{aligned}\right. \end{aligned}$$
(2.2)

from (2.2),we get

$$\begin{aligned} \int _{0}^{t}\left<\frac{d^{2}u_{n}}{dt^{2}}, v\right>+2\alpha\left<\frac{du_{n}}{dt}, v\right>d\tau =\int _{0}^{t}<\triangle u_{n}-|u_{n}|^{p-1}u_{n}+f, v>_{H}d\tau , \end{aligned}$$
(2.3)

it is easily to know

$$\begin{aligned} \left\{ \ \begin{aligned}&\left<\frac{du_{n}}{dt}, v\right>_{H}+2\alpha<u_{n}, v>_{H}= & {} \int _{0}^{t}<\triangle u_{n}-|u|_{n}^{p-1}u_{n}+f, v>_{H}d\tau \\&+2\alpha<\varphi _{n}, v>_{H}+<\psi _{n}, v>_{H},\\&u_{n}(0)=\varphi _{n}, ~~u'_{n}(0)=\psi _{n}.\\ \end{aligned}\right. \end{aligned}$$
(2.4)

for any \(v\in H_n\) is hold, and also for any \(v\in {\widetilde{H}}_n\) ,we have

$$\begin{aligned} \int _{0}^{t}\left<\frac{d^{2}u_{n}}{dt^{2}}, v\right>_{H}+2\alpha \left<\frac{du_{n}}{dt}, v \right>_{H} d\tau =\int _{0}^{t}\left<\triangle u_{n}-|u_{n}|^{p-1}u_{n}+f, v\right>_{H}d\tau . \end{aligned}$$
(2.5)

we can set \(v=\frac{du_n}{dt}\) and put it into (2.5), it yields

$$\begin{aligned} \int _{0}^{t}\int _{\Omega }\left[ \frac{d^{2}u_{n}}{dt^{2}}\frac{du_{n}}{dt}+ 2\alpha \frac{du_{n}}{dt}\frac{du_{n}}{dt}\right] dxd\tau -\int _{0}^{t}\int _{\Omega } \left[ \triangle u_{n}-|u_{n}|^{p-1}u_{n}+f\right] \frac{du_{n}}{dt}dxd\tau =0, \end{aligned}$$

i.e.

$$\begin{aligned} \begin{aligned}&\int _{0}^{t}\left[ \frac{1}{2}\frac{d}{dt}\left<\frac{du_{n}}{dt}, \frac{du_{n}}{dt}\right>_{H_{1}}d\tau +2\alpha \left<\frac{du_{n}}{dt}, \frac{du_{n}}{dt}\right>_{H_{1}}\right. \\&\left. -\left<\triangle u_{n}-|u_{n}|^{p-1}u_{n}, \frac{du_{n}}{dt}\right>\right] d\tau = \int _{0}^{t}\left<f, \frac{du_{n}}{dt}\right>d\tau .\\ \end{aligned} \end{aligned}$$
(2.6)

there exists a \(C^1\) functional \(F: X_2\rightarrow R^1\), such that

$$\begin{aligned} \left<\triangle u_{n}-|u|^{p-1}u_n, \frac{du_n}{dt}\right>=\left<-DF(u_n), \frac{du_n}{dt}\right>, ~~\forall u_n\in X. \end{aligned}$$
(2.7)

where

$$\begin{aligned} F(u)=\int _{\Omega }\left[ \frac{1}{2}|\nabla u|^{2}+\frac{1}{p+1}|u|^{p+1}\right] dx, \end{aligned}$$
(2.8)

and satisfied

$$\begin{aligned} F(u)\rightarrow \infty , \Leftrightarrow \Vert u\Vert _{X_{2}}\rightarrow \infty . \end{aligned}$$
(2.9)

put (2.7) into (2.6),we have

$$\begin{aligned} \begin{aligned}&\int _{0}^{t}\left[ \frac{1}{2}\frac{d}{dt}\left<\frac{du_{n}}{dt}, \frac{du_{n}}{dt}\right>_{H_{1}}d\tau +2\alpha \left<\frac{du_{n}}{dt}, \frac{du_{n}}{dt}\right>_{H_{1}}\right. \\&+\int _{0}^{t}\left<DF(u_{n}, \frac{du_n}{dt})\right>d\tau =\int _{0}^{t}\left<f,\frac{du_n}{dt}\right>d\tau .\\ \end{aligned} \end{aligned}$$
(2.10)

by computing (2.10),we get

$$\begin{aligned} \frac{1}{2}\Vert \frac{du_n}{dt}\Vert _{H_1}^{2}-\frac{1}{2}\Vert \psi _n\Vert _{H_1}^{2} +2\alpha \int _{0}^{t}\left\| \frac{du_n}{dt}\right\| _{H_1}^{2}d\tau \\ +F(u_n)-F(\varphi _n)= \int _{0}^{t}\left<f, \frac{du_n}{dt}\right>d\tau , \end{aligned}$$

i.e.

$$\begin{aligned} \begin{aligned}&F(u_n)+\frac{1}{2}\Vert u'_n\Vert _{H_1}^{2}+2\alpha \int _{0}^{t}\Vert u'_{n}\Vert _{H_1}^2d\tau \\&=\int _{0}^{t}\left<f, \frac{du_n}{dt}\right>d\tau +F(\varphi _n)+\frac{1}{2}\Vert \psi _n\Vert _{H_1}^{2}.\\ \end{aligned} \end{aligned}$$
(2.11)

Now, we deal with the nonlinear and invariational source under the growth condition (1.2) to get

$$\begin{aligned} \begin{aligned} \left|\left<f, \frac{du_n}{dt}\right>\right|&=\left|\int _{\Omega }f(u_n,\nabla u_n,x,t)\frac{du_n}{dt}dx\right|&\\&\le \int _{\Omega }|f(u_{n},\nabla u_n,x,t)|\left|\frac{du_n}{dt}\right|dx&\\&\le \frac{1}{2}\int _{\Omega }\left|\frac{du_n}{dt}\right|^{2}+\frac{1}{2} \int _{\Omega }|f(u_{n},\nabla u_n,x,t)|^2dx&\\&\le \frac{1}{2}\int _{\Omega }\left|\frac{du_n}{dt}\right|^{2}dx+C\int _{\Omega } [|\nabla u_{n}|^{2s}+|u_n|^{2q}+g^{2}(x,t)]dx&\\&\le \frac{1}{2}\Vert u'_{n}\Vert _{H_{1}}^{2}+C_{1}\int _{\Omega } \left[ \frac{1}{2}|\nabla u_n|^{2}+\frac{1}{p+1}|u_n|^{p+1}\right] dx+C_{2}&\\&=\frac{1}{2}\Vert u'_n\Vert _{H_1}^{2}+C_{1}F(u_n)+C_{2},&\end{aligned} \end{aligned}$$
(2.12)

where \(C,C_1,C_2\in R^+\) and \(C_1>2C, q\le \frac{p+1}{2},s\le 1,\) put (2.12) into (2.11), we can conclude that

$$\begin{aligned} F(u_n)+\frac{1}{2}\Vert u'_n\Vert _{H_{1}}^{2}\le C_{4}\int _{0}^{t}\left[ F(u_n)+\frac{1}{2}\Vert u'_{n}\Vert _{H_{1}}^{2}\right] d\tau +h(t), \end{aligned}$$
(2.13)

where

$$\begin{aligned} h(t)=C|\Omega |\int _{0}^{t}g^{2}(\tau )d\tau +\frac{1}{2}\Vert \psi _n\Vert _{H_{1}}^{2} +\sup _{n}F(\varphi _n), C_{4}>0, \end{aligned}$$

combine the Gronwall Inequality and (2.13), we get

$$\begin{aligned} F(u_n)+\frac{1}{2}\Vert u'_n\Vert _{H_1}^{2}\le h(0)e^{C_{4}t}+\int _{0}^{t}h'(\tau )e^{C_{4}(t-\tau )}d\tau . \end{aligned}$$
(2.14)

Set \(\varphi \in H_2\), since the \(\triangle\) operator is linear, \({e_n}\) is also the orthogonal basis of \(H_1\),since the weak convergence, we have

$$\begin{aligned} \varphi _n\rightarrow \varphi ~~in ~~H_2, ~\psi _n\rightarrow \psi ~~in ~~H_1, \end{aligned}$$

obviously, \(H_2\hookrightarrow X_2\), then

$$\begin{aligned} \varphi _n\rightarrow \varphi ~~in ~~X_2, ~\psi _n\rightarrow \psi ~~in ~~H_1, \end{aligned}$$
(2.15)

From (2.9), (2.14) and (2.15), we can conclude that for any \(0<T<\infty\),

$$\begin{aligned} \{u_n\}\subset W^{1,\infty }(0,T; H_{1})\cap L^{\infty }(0,T; X_{2})~~ is~~bounded. \end{aligned}$$

Set \(u_n\rightarrow *u_0~~in ~~W^{1,\infty }(0,T; H_{1})\cap L^{\infty }(0,T; X_{2})\), we have

$$\begin{aligned} u_n\rightharpoonup u_0~~in~~W_{loc}^{1,\infty }(0,\infty ;H_1), \\ u_n\rightarrow *u_0 ~~in~~L_{loc}^{\infty }(0,\infty ;X_2). \end{aligned}$$

and

$$\begin{aligned} G:H^{2}(\Omega )\cap L^{p+1}(\Omega )\rightarrow L^{2}(\Omega )~~is~~weakly~~continuous,~~i.e.~~ \\ \lim _{n\rightarrow \infty }\left<Gu_n, v\right>=\left<Gu_0, v\right>, \forall v\in L^{2}(\Omega ), t\ge 0, \end{aligned}$$

Hence, from (2.4) we conclude that \(u_0\) is a global weak solution of the Eq. (1.1).

The proof is completed.

Remark 2.1

In Theorem 2.1, the gradient source f satisfies the growth condition (1.2)in order to ensure

$$\begin{aligned} \left|\left<f,\frac{du_{n}}{dt}\right>\right|\le \frac{1}{2}\Vert u'_{n}\Vert _{H_{1}}^{2}+C_{1}F(u_n)+C_2. \end{aligned}$$
(2.16)

we can give a further weak growth condition from (2.11) as follows

$$\begin{aligned} \left|\int _{0}^{t}\left<f,\frac{du_{n}}{dt}\right>d\tau \right|\le C\int _{0}^{t}[F(u_n)+\Vert u'_{n}\Vert _{H_{1}}^{2}+g(\tau )]d\tau +aF(u_n)+b, \end{aligned}$$
(2.17)

where \(g\in L_{loc}^1(0,\infty ), 0<a<1,b>0.\)

Now, we discuss the uniqueness of global weak solution for the Eq. (1.1) as follows:

Firstly, we can get the expression of the solution \(u,u_t\) for the Eq. (1.1) from semigroup theory

$$\begin{aligned} \begin{aligned} u=&e^{-\alpha t}[\cos t {(-\mathcal{L})}^{\frac{1}{2}}\varphi +\alpha {(-\mathcal{L})}^{-\frac{1}{2}}\sin t {(-\mathcal{L})}^{\frac{1}{2}}\varphi + {(-\mathcal{L})}^{-\frac{1}{2}}\sin t {(-\mathcal{L})}^{\frac{1}{2}}\psi \\&+\int _{0}^{t}e^{-\alpha (t-\tau )} {(-\mathcal{L})}^{-\frac{1}{2}}\sin (t-\tau ) {(-\mathcal{L})}^{\frac{1}{2}}(-|u|^{p-1}u+f)]d\tau ,\\ \end{aligned} \end{aligned}$$
(2.18)
$$\begin{aligned} \begin{aligned} u_t=&-\alpha u+e^{-\alpha t}[- {(-\mathcal{L})}^{\frac{1}{2}}\sin t {(-\mathcal{L})}^{\frac{1}{2}}\varphi +\alpha \cos t {(-\mathcal{L})}^{\frac{1}{2}}\varphi \\&+\cos t {(-\mathcal{L})}^{\frac{1}{2}}\psi +\int _{0}^{t}e^{-\alpha (t-\tau )}\cos (t-\tau ) {(-\mathcal{L})}^{\frac{1}{2}}(-|u|^{p-1}u+f)]d\tau . \end{aligned} \end{aligned}$$
(2.19)

where \({\mathcal {L}}=\triangle +\alpha ^2 I:X_1\rightarrow X\) is a sectorial operator.

We have the uniqueness theorem as follows

Theorem 2.2

(Uniqueness) If \(u\in W^{1,\infty }(0,T; L^{2}(\Omega )) \cap L^{\infty }_{loc}(0,T;H^{1}_{0}(\Omega )\cap L^{p+1}(\Omega ))\) is a global weak solution of Eq. (1.1), and \(g:H^{1}_{0}(\Omega )\cap L^{p+1}(\Omega ))\rightarrow L^{2}(\Omega )\) is \(C^1\), then the solution u is unique.

where \(g(u)=-|u|^{p-1}u+f(u,\nabla u,x,t).\)

Proof

Set \(u_1,u_2\in W^{1,\infty }(0,T;L^{2}(\Omega )) \cap L^{\infty }_{loc}(0,T; H^{1}_{0}(\Omega )\cap L^{p+1}(\Omega ))\) are the solutions of the Eq. (1.1), then we get \(u_i\in C^{0}((0,T),H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega )), i=1,2,\) and

$$\begin{aligned} \begin{aligned}&\Vert u_{1}-u_{2}\Vert _{X_{\frac{1}{2}}}&\\&=\Vert (-\triangle )^{\frac{1}{2}} (u_{1}-u_{2})\Vert _{H_1}&\\&\le C\int _{0}^{t}\Vert [|u_{2}|^{p-1}u_{2}-|u_1|^{p-1}u_1] +[f(u_1,\nabla u_{1},x,t)-f(u_2,\nabla u_{2},x,t)]\Vert _{H_1}d\tau&\\&=C\int _{0}^{t}\Vert [|u_{2}|^{p-1}u_{2}-|u_1|^{p-1}u_1]+ [f(u_1,\nabla u_{1},x,t)-f(u_2,\nabla u_{1},x,t)]&\\&+[f(u_2,\nabla u_{1},x,t)-f(u_2,\nabla u_{2},x,t)]\Vert _{H_1}d\tau&\\&\le C\int _{0}^{t}[\Vert |{\widetilde{u}}|^{p-1}(u_1-u_2)+D_{z}f({\widetilde{u}},\nabla u_1,x,t)(u_1-u_2)&\\&+D_{\eta }f(u_2,\nabla {\widetilde{u}},x,t)(\nabla u_1-\nabla u_2)\Vert _{H_1}]d\tau&\\&\le C_1\int _{0}^{t}\Vert u_1-u_2\Vert _{H_1}d\tau +C_2\int _{0}^{t}\Vert \nabla u_1-\nabla u_2\Vert _{H_1}d\tau ,&\end{aligned} \end{aligned}$$
(2.20)

by applying the poinc\(\acute{a}\)re inequality, we can conclude that

$$\begin{aligned} \begin{aligned} \Vert u_{1}-u_{2}\Vert _{X_{\frac{1}{2}}}&\le C_{1}\int _{0}^{t} \Vert u_{1}-u_{2}\Vert _{X_{\frac{1}{2}}}d\tau +C_{2}\int _{0}^{t}\Vert \nabla u_{1}-\nabla u_2\Vert _{X_{\frac{1}{2}}}d\tau&\\&=(C_1+C_2)\int _{0}^{t}\Vert u_{1}-u_{2}\Vert _{X_{\frac{1}{2}}}d\tau . \end{aligned} \end{aligned}$$
(2.21)

it implied that

$$\begin{aligned} \Vert u_{1}-u_{2}\Vert _{X_{\frac{1}{2}}}\le 0, \end{aligned}$$

from Gronwall inequality.

Obviously,

$$\begin{aligned} \Vert u_1-u_2\Vert _{X_{\frac{1}{2}}}\rightarrow 0, \end{aligned}$$

so

$$\begin{aligned} u_1=u_2. \end{aligned}$$

Now, we focus on the existence of strong solution for the Eq. (1.1).

Note \(Gu=\triangle u-|u|^{p-1}u+f(u,\nabla u,x,t)\) and satisfies that

$$\begin{aligned} |<Gu, v>|\le \frac{1}{2}\Vert v\Vert _{H_{1}}^{2}+CF(u)+g(t), \quad ~C>0,~g\in L_{loc}^{1}(0,\infty ), \forall v\in {\widetilde{H}}_{n}, \end{aligned}$$
(2.22)

where \(F(u)=\int _{\Omega }[\frac{1}{2}|\nabla u|^{2}+\frac{1}{p+1}|u|^{p+1}]dx,~H_1=L^2(\Omega ).\)

Theorem 2.3

(Strong Solution) Under condition (2.22), for any \((\varphi , \psi )\in H^2(\Omega )\times H_0^{1}(\Omega )\), then the Eq.(1.1) has a global strong solution

$$\begin{aligned} u\in W_{loc}^{2,2}(0,\infty ;\,L^{2}(\Omega )). \end{aligned}$$

Proof

For any \(v\in {\widetilde{H}}_n\), we can set \(v=\frac{d^2u_n}{dt^2}\) and put it into (2.5),

$$\begin{aligned} \int _{0}^{t}\left[ \left<\frac{d^{2}u_n}{dt^2},\frac{d^{2}u_n}{dt^2}\right>_{H_1}+2\alpha \left<\frac{du_n}{dt},\frac{d^{2}u_n}{dt^2}\right>_{H_1}\right] d\tau =\int _{0}^{t} <Gu_n,\frac{d^{2}u_n}{dt^2}>_{H_1}d\tau , \end{aligned}$$
(2.23)

from (2.22), we get

$$\begin{aligned} \begin{aligned}&\int _{0}^{t}\left<\frac{d^{2}u_n}{dt^2},\frac{d^{2}u_n}{dt^2}\right>_{H_1}d\tau +\gamma \Vert u'_n\Vert _{H_1}^{2}&\\&\le \gamma \Vert \psi _{n}\Vert _{H_{1}}^{2}+\int _{0}^{t}\left[ \frac{1}{2}\left\| \frac{d^{2}u_n}{dt^2}\right\| _{H_{1}}^2+CF(u_n)+g(\tau )\right]d\tau .&\end{aligned} \end{aligned}$$

combine (2.14), it implies that

$$\begin{aligned} \int _{0}^{t}\left\| \frac{d^{2}u_n}{dt^{2}}\right\| _{H_1}^{2}d\tau \le C_1,~(C_1>0). \end{aligned}$$
(2.24)

i.e.,for any \(0<T<\infty\),

$$\begin{aligned} \{u_n\}\subset W^{2,2}(0,T; \, H_1)~~is~~bounded. \end{aligned}$$

since

$$\begin{aligned} G:H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega )\rightarrow L^{2}(\Omega ) \end{aligned}$$

is compact mapping, so

$$\begin{aligned} u\in W_{loc}^{2,2}(0,\infty ;\, H_1). \end{aligned}$$

The proof is completed.

3 Existence of Global Attractor

Next, we introduce the concepts and definitions of invariant sets, global attractors, and \(\omega\)-limit compactness sets for the semigroup S(t).

Definition 3.1

Let S(t) be a semigroup defined on X. One set \(\Sigma \subset X\) is called an invariant set of S(t) if \(S(t)\Sigma = \Sigma , \forall t\ge 0\). An invariant set \(\Sigma\) is an attractor of S(t) if \(\Sigma\) is compact, and there exists a neighborhood \(U\subset X\) of \(\Sigma\) such that for any \(u_{0}\in U\),

$$\begin{aligned} \inf _{v\in \Sigma }\Vert S(t)u_{0}-v\Vert _{X}\rightarrow 0, \ as \ t\rightarrow 0. \end{aligned}$$

In this case, we say that \(\Sigma\) attracts U. Especially, if \(\Sigma\) attracts any bounded set of X, \(\Sigma\) is called a global attractor of S(t) in X.

Definition 3.2

Let X be an infinite dimensional Banach space and A be a bounded subset of X. The measure of noncompactness \(\gamma (A)\) of A is defined by

$$\begin{aligned} \gamma (A)=\inf \{\delta >0\mid A \ exists \ a \ finite \ cover \ by \ sets \ whose \ diameter\le \delta \}. \end{aligned}$$

Lemma 3.1

[16]  If \({A_{n}}\subset X\) is a sequence bounded and closed sets, \(A_{n}\ne \emptyset , A_{n+1}\subset A_{n}\), and \(\gamma (A_{n})\rightarrow 0, (n\rightarrow \infty )\), then the set \(A=\cap _{n=1}^{\infty } A_{n}\) is a nonempty compact set.

Definition 3.3

[17]  A semigroup \(S(t): X\rightarrow X (t\ge 0)\) in X is called \(\omega\)- limit compact, if for any bounded set \(B\subset X\) and \(\forall \varepsilon >0\), there exists \(t_{0}\) such that

$$\begin{aligned} \gamma (\cup _{t\ge t_{0}}S(t)B)\le \varepsilon , \end{aligned}$$

where \(\gamma\) is noncompact measure in X.

For a set \(D\subset X\), we define the \(\omega\)-limit set of D as follows:

$$\begin{aligned} \omega (D)=\bigcap _{s\ge 0}\overline{\bigcup _{t\ge s}S(t)D}, \end{aligned}$$

where the closure is taken in the X-norm.

Lemma 3.2

[15]  Let S(t) be a semigroup in X, then S(t) has a global attractor \({\mathcal {A}}\) in X if and only if

(1) S(t) is \(\omega\)-limit compactness, and

(2) there is a bounded absorbing set \(B\subset X\).

In addition, the \(\omega\)-limit set of B is the attractor \({\mathcal {A}}=\omega (B)\).

Remark 3.1

Although the lemma has been proved partly in [15], we still give a proof here. Our proof is different from that in [27] but is similar to that in [17]. We adopt and present the proof also because we will use the same method to obtain the existence of the global attractor.

Proof

Step1. To prove the sufficiency of Lemma 3.2:

(a). Since S(t) is \(\omega\)-limit compactness, i.e., for any bounded set \(B\subset X\) and \(\forall \varepsilon >0,\) there exists a \(t_{0}\), such that:

$$\begin{aligned} \gamma \left( \bigcup _{t\ge t_{0}}S(t)B\right) \le \varepsilon . \end{aligned}$$

So, we know \(\omega (B)=\bigcap _{t_0= 0}^{\infty }\overline{\bigcup _{t\ge t_{0}}S(t)B}\) is a compact set from the Lemma [2.3].

(b). the nonempty of \(\omega (B)\);

For \(B\ne \emptyset\), so \(\overline{\bigcup _{t\ge s}S(t)B}\ne \emptyset , \forall s\ge 0,\)

and

$$\begin{aligned} \overline{\bigcup _{t\ge s_{1}}S(t)B}\subset \overline{\bigcup _{t\ge s_{2}}S(t)B}, \forall s_{1}\ge s_{2}, \end{aligned}$$

we can obtain

$$\begin{aligned} \omega (B)=\bigcap _{s\ge 0}^{\infty }\overline{\bigcup _{t\ge s}S(t)B}\ne \emptyset . \end{aligned}$$

(c). the invariant of \(\omega (B)\);

For \(x\in \omega (B) \Leftrightarrow\) there exists \(\{x_{n}\}\in B\) and \(t_{n}\rightarrow \infty\), such that \(S(t_{n})x_{n}\rightarrow x.\)

If \(y\in S(t)\omega (B)\), then for some \(x\in \omega (B), y=S(t)x\).

Hence, there exists \(\{x_{n}\}\subset B, t_{n}\rightarrow \infty\), such that

$$\begin{aligned} S(t)S(t_{n})x_{n}=S(t+t_{n})x_{n}\rightarrow S(t)x=y. \end{aligned}$$

In conclusion, \(y\in \omega (B), S(t)\omega (B)\in \omega (B), \forall t\ge 0.\)

If \(x\in \omega (B)\), fix \(\{x_{n}\}\subset B\) and \({t_{n}}\), such that

$$\begin{aligned} S(t)x_{n}\rightarrow x, \ as \ t_{n}\rightarrow \infty , n\rightarrow \infty . \end{aligned}$$

Since S(t) is \(\omega\)-limit compact, i.e., there exists a \(y\in H\), such that

$$\begin{aligned} S(t)\bigcap _{t_{n}\ge 0}\overline{\bigcup _{t\ge t_{n}}S(t_{n})x_{n}}\rightarrow y, \ n\rightarrow \infty . \end{aligned}$$

so \(y\in \omega (B)\).

For

$$\begin{aligned} \bigcap _{t_{n}\ge 0}\overline{\bigcup _{t\ge t_{n}}S(t_{n})x_{n}} =\bigcap _{t_{n}\ge 0}\overline{\bigcup _{t\ge t_{n}}S(t)S(t_{n}-t)x_{n}}\rightarrow \bigcap _{t_{n}\ge 0}\overline{\bigcup _{t\ge t_{n}}S(t)y} \end{aligned}$$

and

$$\begin{aligned} S(t_{n})x_{n}\rightarrow x\in \omega (B), \end{aligned}$$

it implies that

$$\begin{aligned} S(t)y\rightarrow x, \ \omega (B)\subset S(t)\omega (B). \end{aligned}$$

In conclusion, combine (a–c) and condition (1.2), step 1 has been proved.

Step 2. To prove the necessary of Lemma 3.2.

If \({{\mathcal {A}}}\) is a global attractor, then the \(\varepsilon\)-neighborhood \(U_{\varepsilon }({{\mathcal {A}}})\subset X\) is a absorbing set. So we need only to prove S(t) is \(\omega\)-limit compactness.

Since \(U_{\varepsilon }({{\mathcal {A}}})\) is a absorbing set, for any bounded set \(B\subset X\) and \(\varepsilon > 0,\) there exists a time \(t_{\varepsilon }(B)> 0\) such that

$$\begin{aligned} \bigcup _{t\ge t_{\varepsilon }(B)}S(t)B\subset U_{\frac{\varepsilon }{4}}({\mathcal {A}})=\{x\in X| dist(x,{\mathcal {A}})< \frac{\varepsilon }{4}\}. \end{aligned}$$

On the other hand, \({\mathcal {A}}\) is a compact set, there exists finite element \(x_{1}, x_{2}, \cdots , x_{n} \in X\) such that

$$\begin{aligned} {\mathcal {A}}\subset \bigcup _{k=1}^{n}U(x_{k}, \frac{\varepsilon }{4}). \end{aligned}$$

Then,

$$\begin{aligned} U_{\frac{\varepsilon }{2}}({\mathcal {A}})\subset \bigcup _{k=1}^{n}U(x_{k}, \frac{\varepsilon }{2}), \end{aligned}$$

it implies that

$$\begin{aligned} \gamma \left(\bigcup _{t\ge t_{\varepsilon }(B)}S(t)B\right)\le \gamma \left(U_{\frac{\varepsilon }{4}}({\mathcal {A}})\right)\le \varepsilon . \end{aligned}$$

Hence, the Lemma 3.2. has been proved.

Fix the spaces as follows

$$\begin{aligned} X_2=X_1=H_{0}^{1}(\Omega )\cap L^{p+1}(\Omega ), X=C_{0}^{\infty }(\Omega ), \\ H_{2}=H^{2}(\Omega ), H_{1}=L^{2}(\Omega ), H=X_{2}\times H_1. \end{aligned}$$

Assume that

$$\begin{aligned} |\Omega |^{\frac{1}{2q}-\frac{1}{2p}}\le \frac{1}{2}, ~p>1, q\le \frac{p+1}{2}. \end{aligned}$$

Theorem 3.1

For any \((\varphi , \psi )\in (H_{0}^{1}(\Omega )\times L^{2}(\Omega ))\), the gradient source f satisfies the growth restriction (1.2), the domain \(\Omega\) satisfies the condition \(|\Omega |^{\frac{1}{2q}-\frac{1}{2p}}\)and the exponents of p satisfies \(1<p<\frac{n}{n-2}, n\ge 3\) or \(1<p<\infty , n=1,2\), then the problem (1.1) has a global attractor \({\mathcal {A}}\) in \((H_{0}^{1}(\Omega )\times L^{2}(\Omega ))\).

Remark 3.2

Comparing the Remark 3.1., we divide the operator G(u) of Definition (2.1)into two parts: L and T, where L is a linear operator while T is a nonlinear operator. We obtain the global attractor of the problem (1.1) by using the Lemma 2.4.

Proof

According to Lemma 3.5., we prove Theorem 3.1 in the following three steps.

Step 1. The problem (1.1) has a globally unique weak solution;

Step 2. To prove S(t) has a bounded absorbing set in \(H_{0}^{1}(\Omega )\times L^{2}(\Omega );\)

Step 3. S(t) is \(\omega\)-limit compactness.

From the Theorems 2.1 and 2.2, we get the problem (1.1) has a global unique weak solution \((u, u_{t})\in C^{0}(0,\infty X_2\times H_1)\). Equation (1.1) generates a semigroup:

$$\begin{aligned} S(t): X_2\times H_1\rightarrow X_2\times H_1 \end{aligned}$$

Firstly, we demonstrate S(t) has a bounded absorbing set in \(H_{0}^{1}(\Omega )\times L^{2}(\Omega ):\)

The Eq. (1.1) is equivalent to the parabolic equations as follows

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{du}{\partial t}=-\alpha u+v,\\ \frac{dv}{\partial t}=\triangle u+\alpha ^{2}u-\alpha v-|u|^{p-1}u+f(u,\nabla u,x,t), \end{array} \right. \end{aligned}$$
(3.1)

Multiply (3.1) by \((-\triangle u, v)\) and take the inner product in \(H_1\), we have

$$\begin{aligned} \left<\frac{du}{dt},-\triangle u\right>_{H_1}= & {} -\alpha<u,-\triangle u>_{H_1}+<-\triangle u,v>_{H_1}, \end{aligned}$$
(3.2)
$$\begin{aligned} \left<\frac{dv}{dt},v\right>_{H_1}= & {} <\triangle u,v>_{H_1}+<\alpha ^{2}u,v>_{H_1}-<\alpha v+|u|^{p-1}u-f,v>_{H_1}, \end{aligned}$$
(3.3)

(3.2)+(3.3),

$$\begin{aligned} \begin{aligned}&\left<\frac{du}{dt},-\triangle u\right>_{H_1}+\left<\frac{dv}{dt},v\right>_{H_1}&\\&=- \alpha<u,-\triangle u>_{H_1}-\alpha<v,v>_{H_1}+\alpha ^{2}<u,v>_{H_1}&\\&+<-|u|^{p-1}u+f,v>_{H_1}.&\end{aligned} \end{aligned}$$
(3.4)

moreover,

$$\begin{aligned}<-\triangle u,\omega>_{H_1}=\left<\left( -{\triangle }^{\frac{1}{2}}\right) u, \left( -{\triangle }^{\frac{1}{2}}\right) \omega \right>_{H_1},~\forall u,\omega \in X_2. \end{aligned}$$
(3.5)

integrate (3.4) over [0,t], we can conclude that

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\Vert u\Vert _{X_2}^{2}+\frac{1}{2}\Vert v\Vert _{H_1}^{2}-\frac{1}{2} \Vert \varphi \Vert _{X_{2}}^{2}-\frac{1}{2}\Vert \psi \Vert _{H_{1}}^{2}\\&=\int _{0}^{t}\left[\left<\frac{du}{dt},-\triangle u\right>_{H_{1}}+\left<\frac{dv}{dt},v\right>_{H_1}\right]d\tau \\&=-\alpha \int _{0}^{t}[<u,-\triangle u>_{H_1}+<v,v>_{H_1}-\alpha<u,v>_{H_1}]d\tau \\& \quad +\int _{0}^{t}<-|u|^{p-1}u+f,v>_{H_1}d\tau \\&=-\alpha \int _{0}^{t}\left[\left<{(-\triangle )}^{\frac{1}{2}}u,{(-\triangle )}^{\frac{1}{2}}u\right>_{H_1}+\Vert v\Vert _{H_1}-\alpha<u,v>_{H_1}\right]d\tau\\& \quad +\int _{0}^{t}<-|u|^{p-1}u+f,v>_{H_1}d\tau\\=-\alpha \int _{0}^{t}\left[\Vert u\Vert _{X_{2}}^{2}+\Vert v\Vert _{H_1}^{2}-\alpha<u,v>_{H_1}+<|u|^{p-1} u,v>_{H_1}\right]d\tau\\&+\int _{0}^{t}<f,v>_{H_1}d\tau .&\end{aligned} \end{aligned}$$
(3.6)

from the growth condition (1.2), we have

$$\begin{aligned} \begin{aligned} |<f,v>|_{H_1}&=\left|\int _{\Omega }f(x,t,u,\nabla u)v\right|&\\&\le \int _{\Omega }|f(x,t,u,\nabla u)||v|dx&\\&\le \frac{1}{2}\int _{\Omega }|f(x,t,u,\nabla u)|^{2}dx+\frac{1}{2}\int _{\Omega }|v|^{2}dx&\\&\le \frac{1}{2}\Vert v\Vert _{H_1}^{2}+C\int _{\Omega }[|\nabla u|^{2\gamma } +|u|^{2q}+g^{2}]dx.&\end{aligned} \end{aligned}$$
(3.7)

and

$$\begin{aligned} \begin{aligned}&<|u|^{p-1}u,v>_{H_1}-\Vert u\Vert _{L^{2q}}^{2q}&\\&\le \int _{\Omega }|u|^{p}|v|dx-\Vert u\Vert _{L^{2q}}^{2q}&\\&\le \frac{1}{2}\int _{\Omega }|u|^{2p}dx+\frac{1}{2}\int _{\Omega }|v|^{2}dx -\Vert u\Vert _{L^{2q}}^{2q}&\\&=\frac{1}{2}\Vert u\Vert _{L^{2p}}^{2p}-\Vert u\Vert _{L^{2q}}^{2q}+\frac{1}{2}\Vert v\Vert _{L^{2}}^{2}.&\end{aligned} \end{aligned}$$
(3.8)

since

$$\begin{aligned} 2q\le p+1\le 2p, p\ge 1, \end{aligned}$$

so we have

$$\begin{aligned} L^{2p}\hookrightarrow L^{p+1}\hookrightarrow L^{2q}, \end{aligned}$$

i.e.

$$\begin{aligned} \Vert u\Vert _{L^{2q}}\le C_1\Vert u\Vert _{L^{2p}},~~\left( C_1=|\Omega |^{\frac{1}{2q}-\frac{1}{2p}}\right) . \end{aligned}$$
(3.9)

combine (3.7), (3.8)and (3.9), we make (3.6) be as follows

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\Vert u\Vert _{X_2}^{2}+\frac{1}{2}\Vert v\Vert _{H_1}^{2}-\frac{1}{2} \Vert \varphi \Vert _{X_{2}}^{2}-\frac{1}{2}\Vert \psi \Vert _{H_{1}}^{2}&\\&\le -\alpha \int _{0}^{t}[\Vert u\Vert _{X_2}^{2}+\Vert v\Vert _{H_1}^{2}] d\tau +\int _{0}^{t}\int _{\Omega }(|\nabla u|^{2\gamma }+g^2)dxd\tau ,&\end{aligned} \end{aligned}$$
(3.10)

so

$$\begin{aligned} \begin{aligned}&\Vert u\Vert _{X_2}^{2}+\Vert v\Vert _{H_1}^{2}&\\&\le -\alpha \int _{0}^{t}[\Vert u\Vert _{X_2}^{2}+\Vert v\Vert _{H_1}^{2}] d\tau +h(\varphi ,\psi )+Ct,~~(\gamma \le 1)&\end{aligned} \end{aligned}$$

where \(C>0\). We can conclude that as follows from Gronwall inequality

$$\begin{aligned} \Vert u\Vert _{X_2}^{2}+\Vert v\Vert _{H_1}^{2}\le h(\varphi ,\psi )e^{-kt}+C(1-e^{-t}). \end{aligned}$$
(3.11)

it implies that S(t) has a bounded absorbing set in \(H_{0}^{1}(\Omega )\times L^{2}(\Omega ).\)

At last, we prove S(t) is \(\omega\)-limit compactness.

From the formula in Lemma 2.2, the solution of problem (1.1) can be expressed as follows:

$$\begin{aligned} \begin{aligned} u& =e^{-\gamma t}[\cos t (-\triangle )^{\frac{1}{2}}\varphi +\gamma (-\triangle )^{-\frac{1}{2}}\sin t(-\triangle )^{\frac{1}{2}}\varphi +(-\triangle )^{-\frac{1}{2}}\sin t(-\triangle )^{\frac{1}{2}}\psi ]\\&\quad +\int _{0}^{t}[e^{-\gamma (t-\tau )}(-\triangle )^{-\frac{1}{2}}\sin (t-\tau ) (-\triangle )^{\frac{1}{2}}(-|u|^{p-1}u+f)]d\tau ,\\ \end{aligned} \end{aligned}$$
(3.12)
$$\begin{aligned} \begin{aligned} u_t&= -\gamma u+e^{-\gamma t}[-(-\triangle )^{\frac{1}{2}}\sin t(-\triangle )^{\frac{1}{2}}\varphi +\gamma \cos t(-\triangle )^{\frac{1}{2}}\varphi \\& \quad +\cos t(-\triangle )^{\frac{1}{2}}\psi ]+\int _{0}^{t}e^{-\gamma (t-\tau )}\cos (t-\tau ) (-\triangle )^{\frac{1}{2}}(-|u|^{p-1}u+f)]d\tau . \end{aligned} \end{aligned}$$
(3.13)

Since the linear operator

$$\begin{aligned} \triangle : H^{2}(\Omega )\times H_{0}^{1}(\Omega )\rightarrow L^{2}(\Omega ) \end{aligned}$$

is symmetrical sector operator, it has eigenvalue sequence:

$$\begin{aligned} 0> \lambda _{1}\ge \lambda _{2}\ge \cdots , \ \lambda _{k}\rightarrow -\infty , k\rightarrow \infty . \end{aligned}$$

Then

$$\begin{aligned} \sin t(-\triangle )^{\frac{1}{2}}v= & {} \sum _{j=1}^{\infty }v_{j}\sin \sqrt{-\lambda _{j}}te_{j}, \end{aligned}$$
(3.14)
$$\begin{aligned} \cos t(-\triangle )^{\frac{1}{2}}v= & {} \sum _{j=1}^{\infty }v_{j} \cos \sqrt{-\lambda _{j}}te_{j}. \end{aligned}$$
(3.15)

For any \(v=\sum _{j=1}^{\infty }v_{j}e_{j}\in L^{2}(\Omega )\) and \(-\lambda _{j}>0, (j\ge 1),\) operator

$$\begin{aligned} \sin t(-\triangle )^{\frac{1}{2}}, \ \cos t(-\triangle )^{\frac{1}{2}}: \ L^{2}(\Omega )\rightarrow L^{2}(\Omega ) \end{aligned}$$

is uniformly bounded, i.e.

$$\begin{aligned} \Vert \sin t(-\triangle )^{\frac{1}{2}}\Vert _{L^{2}}, \ \Vert \cos t(-\triangle )^{\frac{1}{2}}\Vert _{L^{2}}\le 1, \ \forall t\ge 0. \end{aligned}$$
(3.16)

Furthermore, \((u, u_{t})\) contains two parts:

degenerative term

$$\begin{aligned}&\left[ {\begin{array}{*{20}{c}} {{u^{1}}}\\ {{u_{t}^{1}}} \end{array}} \right] \end{aligned}$$
(3.17)
$$\begin{aligned}&=e^{-\gamma t}\left[ {\begin{array}{*{20}{c}} {{\cos {(-\triangle )}^{\frac{1}{2}}+\gamma {(-\triangle )}^{-\frac{1}{2}} \sin t{(-\triangle )}^{\frac{1}{2}}}} &{} {{{(-\triangle )}^{-\frac{1}{2}}\sin t{(-\triangle )}^{\frac{1}{2}}}}\\ {{\gamma \cos t{(-\triangle )}^{\frac{1}{2}}-{(-\triangle )}^{\frac{1}{2}}\sin t{(-\triangle )}^{\frac{1}{2}}}} &{} {{\cos t{(-\triangle )}^{\frac{1}{2}}}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{\varphi }}\\ {{\psi }} \end{array}} \right] \end{aligned}$$
(3.18)

integral term

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {{u^{2}}}\\ {{u_{t}^{2}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\int _{0}^{t}e^{-\gamma (t-\tau )}{(-\triangle )}^{-\frac{1}{2}} \sin (t-\tau ){(-\triangle )}^{\frac{1}{2}}(-|u|^{p-1}u+f)d\tau }}\\ {{\int _{0}^{t}e^{-\gamma (t-\tau )} \cos (t-\tau ){(-\triangle )}^{\frac{1}{2}}(-|u|^{p-1}u+f)d\tau }} \end{array}} \right] \end{aligned}$$
(3.19)

From the uniformly bounded condition (3.16), we get

$$\begin{aligned} \lim _{t\rightarrow \infty }(u_{1}, u_{t}^{1})=0 \ in \ H_{0}^{1}(\Omega )\times L^{2}(\Omega ); \end{aligned}$$
(3.20)

and for any \((\varphi , \psi )\in B,\)

$$\begin{aligned} \bigcup _{t\ge 0}(u^{2}, u_{t}^{2}) \ is \ compact \ set \ in \ H_{0}^{1}(\Omega )\times L^{2}(\Omega ), \end{aligned}$$
(3.21)

where \(B\subset H_{0}^{1}(\Omega )\times L^{2}(\Omega )\) is a bounded set.

From the growth condition (1.2) and \(H_{0}^{1}(\Omega )\hookrightarrow L^{2p}(\Omega ), (p<\frac{n}{n-2})\), we get

$$\begin{aligned} T: H_{0}^{1}(\Omega )\rightarrow L^{2}(\Omega ) \ is \ compact \ map, \end{aligned}$$

where \(Tu=-|u|^{p-1}u+f(u,\nabla u,x,t).\)

Hence, combine (3.20) and (3.21), for the noncompact measure \(\gamma\) we get

$$\begin{aligned} \begin{aligned}&\gamma \left( \bigcup _{t\ge t_{0}}S(t)B\right)&\\&=\gamma \left( \bigcup _{t\ge t_{0}}(u(t, B),u_{t}(t, B))\right)&\\&\le \gamma \left( \bigcup _{t\ge t_{0}}(u^{1},-\gamma u^{1}+u_{t}^{1})\right) +\gamma \left( \bigcup _{t\ge t_{0}}(u^{2},-\gamma u^{2}+u_{t}^{2})\right)&\\&=\gamma \left( \bigcup _{t\ge t_{0}}(u^{1},-\gamma u^{1}+u_{t}^{1})\right)&\\&\rightarrow 0 ~~~(t_{0}\rightarrow \infty ),&\end{aligned} \end{aligned}$$
(3.22)

it implies that

$$\begin{aligned} S(t)=(u(t, \cdot ), u_{t}(t, \cdot )) \ is \ \omega -limit\ compactness. \end{aligned}$$

At last, combine the step 2 and step 3, applying the Lemma [2.4], the problem (1.1) has a global attractor \({\mathcal {A}}\) in \(H_{0}^{1}(\Omega )\times L^{2}(\Omega ).\)