In this section, we provide the existence results for problem (1.1), and then we show the existence of a global attractor in \(L^{q}(\Omega )\).

### Theorem 3.1

*Assume that* \(2\leq p< N/s\), *M* *satisfies* \((M_{1})\), *g* *satisfies* \((g_{1})\), *and* *f* *fulfils* \((f_{1})\)*–*\((f_{2})\) *with* \(2\leq q< Np/(N-sp)\). *Then problem* (1.1) *admits a unique solution*

$$ u\in C\bigl([0,T];L^{2}(\Omega )\bigr)\cap L^{p}(0,T;W_{0}) \cap L^{q}\bigl(0,T;L^{q}( \Omega )\bigr). $$

*Moreover*, *the mapping* \(u_{0}\rightarrow u(t)\) *is continuous in* \(L^{2}(\Omega )\).

### Proof

The existence of solutions for problem (1.1) can be obtained by using the Galerkin method, see for example [21, 25]. For completeness, we give a sketch of the proof.

Choose a sequence of functions \(\{e_{j}\}_{j=1}^{\infty }\subset C_{0}^{\infty }(\Omega )\) which is an orthonormal basis in \(L^{2}(\Omega )\). We shall find the approximate solutions as follows:

$$\begin{aligned} u_{n}(x,t)=\sum_{j=1}^{n}\bigl( \eta _{n}(t)\bigr)_{j}e_{j}(x)\quad \text{for all } n\in \mathbb{N}, \end{aligned}$$

where the unknown functions \((\eta _{n}(t))_{j}\) are determined by the following ODEs:

$$\begin{aligned} \textstyle\begin{cases} \eta ^{\prime }_{n}(t)=-I_{n}(t,\eta _{n}(t)),\quad t\in \mathbb{R}^{+}, \\ \eta _{n}(0)=U_{0n}. \end{cases}\displaystyle \end{aligned}$$

(3.1)

Here \(U_{0n}= (\int _{\Omega }u_{0n}(x)e_{1}(x)\,dx,\ldots ,\int _{\Omega }u_{0n}(x)e_{n}(x)\,dx )\), \(u_{0n}\rightarrow u_{0} \) in \(W_{0}\),

$$\begin{aligned} \bigl(I_{n}(t,\eta _{n})\bigr)_{j}&=M \bigl([u_{n}]_{s,p}^{p}\bigr)\langle u_{n}, e_{j}\rangle _{W_{0}} + \int _{\Omega }f\bigl(x,u_{n}(x,t)\bigr)e_{j}(x) \,dx \\ &\quad {}- \int _{\Omega }g(x)e_{j}(x)\,dx, \quad j=1,2,\ldots ,n. \end{aligned}$$

The definition of \(\langle u_{n},e_{j}\rangle _{W_{0}}\) is given by

$$ \langle u_{n},e_{j}\rangle _{W _{0}} = \iint _{ \mathcal{Q}} \frac{ \vert u_{n}(x,t)-u_{n}(y,t) \vert ^{p-2} [u_{n}(x,t)-u_{n}(y,t)][e_{j}(x)-e_{j}(y)]}{ \vert x-y \vert ^{N+sp}}\,dx \,dy. $$

By the continuity of *M* and the definition of \(I_{n}\), we know that \(I_{n}\) is continuous on \(\mathbb{R}^{+}_{0}\times \mathbb{R}^{n}\). Then the Peano theorem (see [11]) yields that there exists a local solution of problem (3.1) on \((0, T_{n})\) (\(0< T_{n}<\infty \)). The following a priori estimate implies that the local solution can be extended to \((0,\infty )\).

Multiplying (3.1) by \(\eta _{n}(t)\), we obtain

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{\Omega } \bigl\vert u_{n}(x,t) \bigr\vert ^{2}\,dx+M\bigl([u_{n}]_{s,p}^{p}\bigr) \iint _{\mathcal{Q}}\frac{ \vert u_{n}(x,t)-u_{n}(y,t) \vert ^{p}}{ \vert x-y \vert ^{N+sp}} \,dx \,dy \\ &\quad {}+ \int _{\Omega }f(x,u_{n})u_{n}\,dx= \int _{\Omega }g(x)u_{n}\,dx. \end{aligned}$$

(3.2)

Let \(u_{0}\in W_{0}\cap L^{q}(\Omega )\). Then, multiplying (3.1) by \(\eta _{n}^{\prime }(t)\), we get

$$\begin{aligned} & \int _{\Omega } \biggl\vert \frac{\partial u_{n}(x,t)}{\partial t} \biggr\vert ^{2}\,dx+ \frac{d}{dt} \biggl[\mathscr{M}\bigl([u_{n}]_{s,p}^{p} \bigr) + \int _{\Omega }F(x,u_{n})\,dx \biggr]= \int _{\Omega }g(x)\frac{\partial u_{n}(x,t)}{\partial t}\,dx. \end{aligned}$$

(3.3)

It follows from \((M_{1})\), (1.3), (3.2), and the Hölder inequality that

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{\Omega } \bigl\vert u_{n}(x,t) \bigr\vert ^{2}\,dx+C_{1} \int _{\Omega } \bigl\vert u_{n}(x,t) \bigr\vert ^{q}\,dx\leq C \vert \Omega \vert + \bigl\Vert g(x) \bigr\Vert _{2} \bigl\Vert u_{n}(x,t) \bigr\Vert _{2}. \end{aligned}$$

Further, by \(q\geq 2\) and the Young inequality, we obtain

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \int _{\Omega } \bigl\vert u_{n}(x,t) \bigr\vert ^{2}\,dx + \frac{C_{1}}{2} \int _{\Omega } \bigl\vert u_{n}(x,t) \bigr\vert ^{2}\,dx \leq \biggl[(C+C_{1}) \vert \Omega \vert + \frac{2}{C_{1}} \bigl\Vert g(x) \bigr\Vert _{2}^{2} \biggr], \end{aligned}$$

which implies that

$$\begin{aligned} & \int _{\Omega } \bigl\vert u_{n}(x,t) \bigr\vert ^{2}\,dx \\ &\quad \leq \int _{\Omega } \bigl\vert u_{0n}(x,0) \bigr\vert ^{2}dxe^{-C_{1}t} + \frac{2(C+C_{1}) \vert \Omega \vert }{c_{1}}\bigl(1-e^{-C_{1}t} \bigr) +\frac{4}{C_{1}} \int _{0}^{t} \Vert g \Vert _{2}^{2}e^{C_{1}(\tau -t)}\,d\tau \\ &\quad \leq C, \end{aligned}$$

(3.4)

where \(C>0\) denotes various constants independent of *n* and *t*. This together with (3.2) deduces that the local solution \(u_{n}\) can be extended to \((0,\infty )\).

Then, using a similar discussion as that in [21], we can obtain that the limit of \(\{u_{n}\}\) is a solution of problem (1.1).

Next we prove that problem (1.1) only has one solution. Assume that *u* and *v* are two solutions of problem (1.1). Taking \(\varphi =u-v\) as a test function in Definition 1.1, we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{\Omega } \vert u-v \vert ^{2}\,dx +M \bigl([u]_{s,p}^{p}\bigr) \langle u,u-v\rangle _{W_{0}}-M\bigl([v]_{s,p}^{p}\bigr)\langle u,u-v\rangle _{W_{0}} \\ &\quad {}+ \int _{\Omega }\bigl(f(x,u)-f(x,v)\bigr) (u-v)\,dx=0. \end{aligned}$$

Note that

$$\begin{aligned} \langle u,u-v\rangle _{W_{0}}=[u]_{s,p}^{p}-\langle u,v\rangle _{W_{0}}. \end{aligned}$$

By the Young inequality, we have

$$\begin{aligned} \langle u,v\rangle _{W_{0}} &= \iint _{\mathcal{Q}} \frac{ \vert u(x)-u(y) \vert ^{p-2}(u(x)-u(y))(v(x)-v(y))}{ \vert x-y \vert ^{N+sp}}\,dx \,dy \\ &\leq \biggl(1-\frac{1}{p} \biggr)[u]_{s,p}^{p} + \frac{1}{p}[v]_{s,p}^{p}. \end{aligned}$$

Thus,

$$\begin{aligned} \langle u,u-v\rangle _{W_{0}}\geq \frac{1}{p}\bigl([u]_{s,p}^{p}-[v]_{s,p}^{p} \bigr). \end{aligned}$$

Similarly,

$$\begin{aligned} \langle v,u-v\rangle _{W_{0}}\leq \frac{1}{p}\bigl([u]_{s,p}^{p}-[v]_{s,p}^{p} \bigr). \end{aligned}$$

Using the above inequalities and assumption (1.2), we arrive at the inequality

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{\Omega } \vert u-v \vert ^{2}\,dx + \frac{1}{p} \bigl[M\bigl([u]_{s,p}^{p}\bigr)-M \bigl([v]_{s,p}^{p}\bigr) \bigr]\bigl([u]_{s,p}^{p}-[v]_{s,p}^{p} \bigr) \leq \lambda \int _{\Omega } \vert u-v \vert ^{2}\,dx. \end{aligned}$$

Since *M* is a nondecreasing function, we deduce that

$$ \frac{d}{dt} \int _{\Omega } \vert u-v \vert ^{2}\,dx\leq 2\lambda \int _{\Omega } \vert u-v \vert ^{2}\,dx, $$

which implies that \(u-v=0\) a.e. in \(\Omega \times (0,\infty )\). Hence the solution is unique. With a similar discussion as the uniqueness of solution, we can obtain the continuity of the mapping \(u_{0}\rightarrow u (t)\) in \(L^{2}(\Omega )\). □

Now we define a functional \(E:W_{0}\rightarrow \mathbb{R}\) by

$$\begin{aligned} E(u)=\frac{1}{p}\mathscr{M}\bigl([u]_{s,p}^{p}\bigr)+ \int _{\Omega } F(x,u)\,dx- \int _{\Omega }g(x)u\,dx \end{aligned}$$

for all \(u\in W_{0}\), where \(F(x,u)=\int _{0}^{u} f(x,\xi )\,d\xi \). Then we have the following.

### Lemma 3.1

*Assume that* \(u_{0}\in W_{0}\cap L^{q}(\Omega )\). *Let* *u* *be a solution of problem* (1.1), *then*

$$\begin{aligned} E\bigl(u(x,t)\bigr)=E(u_{0})- \int _{0}^{t} \int _{\Omega } \bigl\vert u_{\tau }(x,\tau ) \bigr\vert ^{2}dxd \tau ,\quad t>0. \end{aligned}$$

(3.5)

### Proof

Let us recall that the solution of problem (1.1) can be obtained as the limit of the following sequence of Galerkin’s approximation (see [25]):

$$\begin{aligned} u_{n}(x,t)=\sum_{j=1}^{n} \bigl(g_{n}(t)\bigr)_{j}e_{j}(x),\quad n=1,2, \ldots , \end{aligned}$$

where \(g_{n}(t)\in C^{1}[0,T]\) and \(\{e_{j}\}\subset C_{0}^{\infty }(\Omega )\) is an orthonormal basis in \(L^{2}(\Omega )\). Let *u* be a sufficiently smooth solution of problem (1.1)(or the approximate solution \(u_{n}\)). Choosing \(\varphi =u_{t}\) in Definition 1.1 and using the fact that

$$\begin{aligned} \langle u,u_{t}\rangle _{W_{0}}=\frac{1}{p} \frac{d}{dt}\mathscr{M}\bigl([u]_{s,p}^{p}\bigr), \end{aligned}$$

we have

$$\begin{aligned} \int _{\Omega } \vert u_{t} \vert ^{2}\,dx+ \frac{d}{dt}E\bigl(u(x,t)\bigr)=0, \end{aligned}$$

which implies that the function \(E(u(x,t))\) is nonincreasing with respect to *t*. Moreover, integrating the above equality with respect to *t* from 0 to *t*, we arrive at the equality

$$\begin{aligned} \int _{0}^{t} \int _{\Omega } \vert u_{\tau } \vert ^{2}\,dx \,d\tau +E\bigl(u(x,t)\bigr)-E(u_{0})=0. \end{aligned}$$

This ends the proof. □

By Theorem 3.1, the solution of problem (1.1) generates a semigroup \(\{S(t)\}_{t\geq 0}\) in \(L^{2}(\Omega )\). Next, we show that the semigroup possesses a global attractor in \(L^{q}(\Omega )\).

### Theorem 3.2

*Under the assumptions of Theorem *3.1, *the semigroup* \(\{S(t)\}_{t\geq 0}\) *associated with problem* (1.1) *possesses an absorbing set in* \(L^{2}(\Omega )\) *and* \(W_{0}\cap L^{q}(\Omega )\), *respectively*.

### Proof

Taking \(\varphi =u\) in Definition 1.1, we obtain

$$ \frac{1}{2}\frac{d}{dt} \int _{\Omega } \vert u \vert ^{2}\,dx+M \bigl([u]_{s,p}^{p}\bigr)[u]_{s,p}^{p}+ \int _{\Omega }f(x,u)u\,dx= \int _{\Omega }gudx. $$

Note that assumption (1.3) implies that

$$ \int _{\Omega }f(x,u)u\,dx\geq c_{1} \int _{\Omega } \vert u \vert ^{q}\,dx-c \vert \Omega \vert . $$

This together with the Young inequality and assumption \((M_{1})\) yields that

$$ \frac{1}{2}\frac{d}{dt} \int _{\Omega } \vert u \vert ^{2} \,dx+m_{0}[u]_{s,p}^{p}+c_{1} \int _{\Omega } \vert u \vert ^{q}\,dx\leq C_{\varepsilon } \int _{\Omega } \vert g \vert ^{2}\,dx+ \varepsilon \int _{\Omega } \vert u \vert ^{2}\,dx+c \vert \Omega \vert . $$

(3.6)

Using the Young inequality, one can deduce that

$$\begin{aligned} \int _{\Omega } \vert u \vert ^{2}\,dx&\leq \int _{\Omega }\frac{2}{q} \vert u \vert ^{q} \,dx + \int _{\Omega }\frac{q-2}{q}\,dx \\ &=\frac{2}{q} \int _{\Omega } \vert u \vert ^{q}\,dx + \frac{q-2}{q} \vert \Omega \vert . \end{aligned}$$

Thus,

$$ \frac{q}{2}c_{1} \int _{\Omega } \vert u \vert ^{2}\,dx \leq c_{1} \int _{\Omega } \vert u \vert ^{q}\,dx+ \frac{q-1}{2}c_{1} \vert \Omega \vert . $$

Inserting this inequality into (3.6), we get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{\Omega } \vert u \vert ^{2} \,dx+m_{0}[u]_{s,p}^{p}+ \frac{q}{2}c_{1} \int _{\Omega } \vert u \vert ^{2}\,dx \\ &\quad \leq \varepsilon ^{-1} \int _{\Omega } \vert g \vert ^{2}\,dx+\varepsilon \int _{ \Omega } \vert u \vert ^{2}\,dx+ \biggl(c+ \frac{q-1}{2}c_{1} \biggr) \vert \Omega \vert . \end{aligned}$$

Choose \(\varepsilon =\frac{qc_{1}}{4}\). Then

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{\Omega } \vert u \vert ^{2}\,dx+ \frac{q c_{1}}{4} \int _{\Omega } \vert u \vert ^{2}\,dx \\ &\quad \leq \frac{4}{qc_{1}} \int _{\Omega } \vert g \vert ^{2}\,dx + \biggl(c+ \frac{q-1}{2}c_{1} \biggr) \vert \Omega \vert . \end{aligned}$$

(3.7)

Then, using a similar discussion as (3.4), we get that there exists \(t_{0}>0\) such that

$$ \bigl\Vert u(x,t) \bigr\Vert _{2}\leq C \quad \text{for any } t \geq t_{0}. $$

Thus, the semigroup has an absorbing set in \(L^{2}(\Omega )\). Integrating (3.6) with respect to *t* over \([t, t+1]\), \(t \geq t_{0}\), we obtain

$$\begin{aligned} & \int ^{t+1}_{t} \bigl(m_{0}\bigl[u(x,\tau )\bigr]_{s,p}^{p}+c_{1} \bigl\Vert u(x,\tau ) \bigr\Vert _{q}^{q} \bigr)\,d\tau \\ &\quad \leq C_{\varepsilon } \Vert g \Vert ^{2}_{2}+ \bigl\Vert u(x,t) \bigr\Vert ^{2}_{2}+C \vert \Omega \vert \leq C \quad \text{for } t\geq t_{0}, \end{aligned}$$

which implies that

$$\begin{aligned} \int ^{t+1}_{t} \bigl(\bigl[u(x,\tau ) \bigr]_{s,p}^{p}+ \bigl\Vert u(x,\tau ) \bigr\Vert _{q}^{q} \bigr)\,d\tau \leq C\quad \text{for } t\geq t_{0}. \end{aligned}$$

(3.8)

On the other hand, using Lemma 3.1 and the Young inequality, we deduce

$$ \frac{1}{2} \int _{\Omega } \vert u_{t} \vert ^{2}\,dx+ \frac{d}{dt} \mathscr{M}\bigl([u]_{s,p}^{p}\bigr) + \frac{d}{dt} \int _{\Omega }F(x,u)\,dx\leq \frac{1}{2} \Vert g \Vert ^{2}_{2}. $$

(3.9)

By assumption (1.3), we have

$$ c_{1} \vert u \vert ^{q}-c\leq F(x,u) \leq c_{2} \vert u \vert ^{q}+c. $$

(3.10)

Integrating (3.9) over \([\tau ,t + 1] \), \(t_{0} \leq t < \tau < t + 1\), one can deduce

$$\begin{aligned} &\mathscr{M}\bigl(\bigl[u(x,t)\bigr]_{s,p}^{p}\bigr) + \int _{\Omega }F\bigl(x,u(x,t)\bigr)\,dx \\ &\quad \leq C \Vert g \Vert ^{2}_{2}+ \biggl(\mathscr{M}\bigl( \bigl[u(x,\tau )\bigr]_{s,p}^{p}\bigr) + \int _{\Omega }F\bigl(x,u(x,\tau )\bigr)\,dx \biggr). \end{aligned}$$

Integrating the above inequality with respect to *τ* between *t* and \(t + 1\), we obtain

$$\begin{aligned} &\mathscr{M}\bigl(\bigl[u(x,t+1)\bigr]_{s,p}^{p}\bigr) + \int _{\Omega }F\bigl(x,u(x,t+1)\bigr)\,dx \\ &\quad \leq C \Vert g \Vert ^{2}_{2} + \int _{t}^{t+1} \biggl(\mathscr{M}\bigl(\bigl[u(x, \tau )\bigr]_{s,p}^{p}\bigr) + \int _{\Omega }F\bigl(x,u(x,\tau )\bigr)\,dx \biggr)\,d\tau . \end{aligned}$$

Gathering (3.8) and (3.10), we get

$$ \bigl[u(x,t)\bigr]_{s,p}^{p}+ \bigl\Vert u(x,t) \bigr\Vert _{q}^{q}\leq C \quad \text{for all } t \geq t_{0}+1. $$

The proof is complete. □

By the compact imbedding results in [35] and [5, Theorem 6.7], we are now in a position to obtain the global attractor in \(L^{q}(\Omega )\).

### Proof of Theorem 1.1

The proof is inspired by [30]. Let \(B_{0}\) be an absorbing set in \(L^{q}(\Omega )\). We define the *ω*-limit set of \(B_{0}\) as

$$\begin{aligned} \omega (B_{0}):=\bigcap_{\tau \geq 0} \overline{ \bigcup_{t\geq \tau }S(t)B_{0}}^{L^{q}(\Omega )}. \end{aligned}$$

Here \(A^{L^{q}(\Omega )}\) denotes the closure of *A* in the topology of \(L^{q}(\Omega )\). Note that \(\varphi \in \omega (B_{0})\) if and only if there exist sequences \(\{\varphi _{n}\}\subset B_{0}\) and \(t_{n}\rightarrow \infty \) such that

$$\begin{aligned} S(t_{n})\varphi _{n}\rightarrow \varphi \quad \text{as } n \rightarrow \infty . \end{aligned}$$

Set \(\mathscr{A}=\omega (B_{0})\). Next we verify that \(\mathscr{A}\) is a global attractor of the semigroup \(S(t)\) in \(L^{q}(\Omega )\).

(1) \(\mathscr{A}\) is compact. Clearly, by the compact imbedding results in [35] and [5, Theorem 6.7], one can obtain that \(\mathscr{A}\) is compact in \(L^{q}(\Omega )\) being \(q\in (2,Np/(N-sp))\).

(2) \(\mathscr{A}\) is invariant. If \(v\in S(t)\mathscr{A}\), then \(v=S(t)\varphi \), \(\varphi \in \mathscr{A}\). Thus, there exist \(\varphi _{n}\) and \(t_{n}\) such that

$$ S(t)S(t_{n})\varphi _{n}=S(t+t_{n})\varphi _{n}\rightarrow S(t) \varphi =v, $$

which implies that \(v\in \mathscr{A}\). If \(v\in \mathscr{A}\), then there exist \(\varphi _{n}\in B_{0}\) and \(t_{n}\rightarrow \infty \) such that \(S(t_{n})\varphi _{n}\rightarrow v\). Observe that, for \(t_{n}\geq t\), the sequence \(S(t_{n}-t)\varphi _{n}\) is compact in \(L^{q}(\Omega )\). Thus, there exist a subsequence \(t_{n_{k}}\rightarrow \infty \) and \(\varphi \in L^{q}(\Omega )\) such that \(S(t_{n_{k}}-t)\varphi _{n_{k}}\rightarrow \varphi \). It follows that \(\varphi \in \mathscr{A}\). By the continuity of \(S(t)\), we deduce

$$\begin{aligned} S(t_{n_{k}})\varphi _{n_{k}}= S(t)S(t_{n_{k}}-t)\varphi _{n_{k}} \rightarrow S(t)\varphi =v. \end{aligned}$$

It yields that \(v\in S(t)\mathscr{A}\). Consequently, we obtain that \(S(t)\mathscr{A}=\mathscr{A}\).

(3) \(\mathscr{A}\) attracts any bounded sets in \(L^{q}(\Omega )\). Arguing by contradiction, we assume that for some bounded set \(B_{1}\) of \(L^{2}(\Omega )\), \({\mathrm{dist}}(S(t)B_{1},\mathscr{A})\) does not limit to 0 as \(t\rightarrow \infty \). Hence there exist \(\delta >0\) and a sequence \(t_{n}\rightarrow \infty \) such that, for all *n*,

$$ {\mathrm{dist}}\bigl(S(t_{n})B_{1},\mathscr{A}\bigr)\geq \delta >0. $$

For each \(n\geq 1\), there exists \(\varphi _{n}\in B_{1}\) such that

$$\begin{aligned} {\mathrm{dist}}\bigl(S(t_{n})\varphi _{n}, \mathscr{A}\bigr)\geq \frac{\delta }{2}>0. \end{aligned}$$

(3.11)

Recall that \(B_{0}\) is an absorbing set. Then \(S(t_{n})\varphi _{n}\subset B_{0}\) for \(t_{n}\geq t_{0}:=t_{0}(B)\). Since \(S(t_{n})\varphi _{n}\) is compact, there exist \(\varphi \in L^{q}(\Omega )\) and a subsequence of \(t_{n}\) denoted by \(t_{n_{k}}\) such that

$$\begin{aligned} \varphi =\lim_{n_{k}\rightarrow \infty } S(t_{n_{k}})\varphi _{n_{k}}= \lim_{n_{k}\rightarrow \infty } S(t_{n_{k}}-t_{0})S(t_{0}) \varphi _{n_{k}}. \end{aligned}$$

It follows from \(S(t_{0})\varphi _{n}\in B_{0}\) that \(\varphi \in \mathscr{A}_{q}\), which contradicts (3.11). □