A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs

Abstract

A class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs are studied. Under suitable assumptions on the exponents of nonlinear source terms and the memory kernel, the existence of global solutions in an appropriate functional space is established by a combination of the theory of potential wells and the Galerkin approximations. Furthermore, the asymptotic behavior of global solutions is obtained by a combination of the theory of potential wells and the perturbed energy method.

1. Introduction

In this paper, we deal with the initial boundary value problem for a class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs.

$$\begin{array}{cc}\hfill & u_{tt}+(a+b{\left[u\right]}_m^{2p-2}){(-\mathsf{\Delta })}^{m}u-{\int }_0^{t}g(t-\tau ){(-\mathsf{\Delta })}^{m}u\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & +u_t={\left|u\right|}^{q-2}u-{\left|u\right|}^{r-2}u,x\in \mathsf{\Omega },t>0,\hfill \end{array}$$

(1)

$$u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),x\in \mathsf{\Omega },$$

(2)

$$u(x,t)=0,x\in {\mathbb{R}}^N\backslash \mathsf{\Omega },t>0,$$

(3)

where ${\left[u\right]}_m$ is the Gagliardo seminorm defined by

$${\left[u\right]}_m:={\left(\int {\int }_{{\mathbb{R}}^{2N}}\frac{{|u(x,t)-u(y,t)|}^2}{{|x-y|}^{N+2m}}\mathrm{d}x\mathrm{d}y\right)}^{\frac{1}{2}},$$

${\left[u\right]}_m^{2p-2}$ means that $2p-2$ is a power of ${\left[u\right]}_m$, and ${(-\mathsf{\Delta })}^m$ is the fractional Laplacian with $0<m<1$, which (up to normalization factors) could be given by

$${(-\mathsf{\Delta })}^m\phi \left(x\right):={\int }_{{\mathbb{R}}^N}\frac{2\phi \left(x\right)-\phi (x+y)-\phi (x-y)}{{\left|y\right|}^{N+2m}}\mathrm{d}y,x\in {\mathbb{R}}^N$$

for all $\phi \in C_0^{\infty }\left({\mathbb{R}}^N\right)$. In addition, $a>0$, $b\ge 0$, $p>1$, $\mathsf{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 1$) is a bounded domain with a Lipschitz boundary, and the functions $u_0\left(x\right)$ and $u_1\left(x\right)$ are specified later.

The problem (1)–(3) can govern the motion of the viscoelastic string with a fractional length (see Ref. [1]). The unknown function $u=u(x,t)$ represents the vertical displacement, $-{\int }_0^{t}g(t-\tau ){(-\mathsf{\Delta })}^{m}u\left(\tau \right)\mathrm{d}\tau$ is the viscoelastic term, $u_t$ is a weakly damped term, ${\left|u\right|}^{q-2}u$ and $-{\left|u\right|}^{r-2}u$ are two nonlinear source terms of different signs, and q, r, and the memory kernel g are introduced later.

In the case where $m=1$, Equation (1) becomes a classical viscoelastic Kirchhoff wave equation. Concerning this class of equations, Torrejón and Yong [2] investigated

$$u_{tt}-{h(\parallel \nabla u\parallel }_2^2)\mathsf{\Delta }u+{\int }_0^{t}g(t-\tau )\mathsf{\Delta }u\left(\tau \right)\mathrm{d}\tau =f,$$

where h is the Kirchhoff function. They obtained existence, uniqueness, and asymptotic behavior of global solutions. Wu and Tsai [3] treated a viscoelastic Kirchhoff wave equation with nonlinear weak damping and source terms:

$$u_{tt}-{h(\parallel \nabla u\parallel }_2^2)\mathsf{\Delta }u+{\int }_0^{t}g(t-\tau )\mathsf{\Delta }u\left(\tau \right)\mathrm{d}\tau +f_2\left(u_t\right)=f_1\left(u\right).$$

They derived existence and blow-up of local solutions, and also established the estimates on the blow-up time. Liu et al. [4] studied a more general viscoelastic Kirchhoff wave equation:

$$u_{tt}-{h(\parallel \nabla u\parallel }_2^2)\mathsf{\Delta }u+{\int }_0^{t}g(t-\tau )\mathsf{\Delta }u\left(\tau \right)\mathrm{d}\tau +f_2(x,t,u,u_t)=f_1(x,t,u).$$

They proved the nonexistence of global solutions.

Fractional partial differential equations arise in continuum mechanics [5], quantum mechanics [6,7], population dynamics [5], anomalous diffusion [8,9], fluid mechanics [10,11], and so on, and they have received considerable attention. Ambrosio and Isernia [12] investigated a fractional stationary Kirchhoff equation:

$$(a+b(1-m){\left[u\right]}_m^2){(-\mathsf{\Delta })}^{m}u=f\left(u\right),$$

where a and b are two parameters. By the minimax arguments, they established the multiplicity of solutions, provided b is sufficiently small. Nyamoradi and Ambrosio [13] dealt with a fractional stationary Kirchhoff equation involving two nonlinear source terms:

$$(a+b{\left[u\right]}_m^{2\theta -2}){(-\mathsf{\Delta })}^{m}u={\lambda f\left(x\right)\left|u\right|}^{q-2}u+{\left|u\right|}^{2_m^{*}-2}u,$$

where $\theta$ and $\lambda$ are also parameters, and $2_m^{*}$ represents the critical exponent of the fractional Sobolev space. Under suitable values of the parameters, they derived the existence and nonexistence of multiple solutions. Fiscella and Mishra [14] studied the following fractional stationary Kirchhoff equation involving two nonlinear source terms:

$$(a+b{\left[u\right]}_m^{2\theta -2}){(-\mathsf{\Delta })}^{m}u-\mu \frac{u}{{\left|x\right|}^{2m}}=\lambda f_1\left(x\right)u^{-p}+f_2\left(x\right)u^{2_m^{*}-1}.$$

They addressed the existence of at least two positive solutions depending on the parameters by exploiting the Nehari manifold. do Ó et al. [15] studied a fractional stationary Kirchhoff equation involving two nonlinear source terms of different signs:

$$(a+b{\left[u\right]}_m^{2\theta -2}){(-\mathsf{\Delta })}^{m}u={\lambda f\left(x\right)\left|u\right|}^{q-2}u-{\left|u\right|}^{2_m^{*}-2}u.$$

By using a variational approach based on the Nehari manifold, they obtained the existence of two positive solutions for suitable values of the parameters. Zhang et al. [16] investigated a fractional stationary Kirchhoff equation involving the nonlocal integro-differential operator:

$$-h\left(\int {\int }_{{\mathbb{R}}^{2N}}{|u\left(x\right)-u\left(y\right)|}^{2}K(x-y)\mathrm{d}x\mathrm{d}y\right){\mathcal{L}}_{K}u=f(x,u),$$

(4)

where $K:{\mathbb{R}}^N\backslash \left\{0\right\}\to (0,\infty )$ is a function satisfying certain assumptions, and ${\mathcal{L}}_K$ denotes the nonlocal integral-differential operator. Equation (4) includes the following fractional Kirchhoff equation as a particular case.

$$h\left({\left[u\right]}_m^2\right){(-\mathsf{\Delta })}^{m}u=f(x,u).$$

By computing the critical groups at zero and at infinity, they derived the existence of at least one nontrivial solution via Morse theory. Molica Bisci and Vilasi [17] considered the following fractional stationary Kirchhoff equation involving two nonlinear source terms:

$$-h\left({\left[u\right]}_m^2\right){\mathcal{L}}_{K}u=\lambda f_1(x,u)+\mu f_2(x,u).$$

By exploiting an abstract critical point theorem for smooth functionals, they obtained the existence of at least three solutions for suitable values of the parameters. Additionally, in the autonomous case, they gave a precise estimate for the range of these parameters by using some properties of the fractional calculus on a specific family of test functions. Concerning the fractional evolution Kirchhoff equations, Xiang et al. [18] studied

$$u_t+h\left({\left[u\right]}_m^2\right){(-\mathsf{\Delta })}^{m}u={\left|u\right|}^{p-2}u.$$

They obtained the existence of non-negative local solutions by using the Galerkin approximations and proved the blow-up of non-negative local solutions with suitable initial data by virtue of a differential inequality technique. Lin et al. [19] studied a fractional evolution Kirchhoff equation of the form

$$u_{tt}+{\left[u\right]}_m^{2\theta -2}{(-\mathsf{\Delta })}^{m}u=f\left(u\right).$$

They utilized the concavity arguments to obtain the blow-up of solutions. Pan et al. [20] considered the following fractional evolution Kirchhoff equation with nonlinear weak damping:

$$u_{tt}+{\left[u\right]}_m^{2\theta -2}{(-\mathsf{\Delta })}^{m}u+|{u_t|}^{r-2}{u}_t+u={\left|u\right|}^{q-2}u.$$

They obtained the global existence, vacuum isolation, asymptotic behavior, and blow-up of solutions by using the theory of potential wells. Recently, Xiang and Hu [21] conducted an investigation on the following fractional viscoelastic Kirchhoff equation:

$$u_{tt}+h\left({\left[u\right]}_m^2\right){(-\mathsf{\Delta })}^{m}u-{\int }_0^{t}g(t-\tau ){(-\mathsf{\Delta })}^{m}u\left(\tau \right)\mathrm{d}\tau +{(-\mathsf{\Delta })}^s{u}_t=\lambda {\left|u\right|}^{q-2}u.$$

They utilized the Galerkin approximations to establish the existence of local and global solutions, and they employed the concavity arguments to derive the blow-up of solutions.

Inspired by the above works, we deal with the problem (1)–(3). There are two features about our results. On the one hand, our results are independent of any parameters. On the other hand, we focus on the effects of the nonlinear source terms of different signs on solutions. From a physical perspective, the two nonlinear source terms of different signs actually represent the two opposing external forces acting on the viscoelastic string with fractional length. In the absence of the external force ${\left|u\right|}^{q-2}u$, it is trivial that the problem (1)–(3) has a global solution with the asymptotic behavior by the arguments similar to Ref. [22]. Our results show that even though the external force ${\left|u\right|}^{q-2}u$ appears, the problem (1)–(3) can still have a global solution with the asymptotic behavior. In addition, our main technical tool is the theory of potential wells that has been widely employed to analyze the qualitative properties of solutions of evolution equations. In this regard, in addition to the work of Ref. [20] mentioned above, we refer to the following work. Xu et al. [23] used the theory of potential wells to investigate the global existence and blow-up of solutions of a sixth-order nonlinear hyperbolic equation. Cavalcanti and Domingos Cavalcanti [24] modified the theory of potential wells to study the global existence and asymptotic behaviour of solutions of a nonlinear evolution equation. Gazzola and Squassina [25] improved the theory of potential wells to obtain the global existence, asymptotic behaviour, and blow-up of solutions of a damped semilinear wave equation. Liu et al. [26] applied the idea of Ref. [25] to deal with a fourth-order damped nonlinear hyperbolic equation. Xu and Su [27], Luo et al. [28], and Liu and Li [29] introduced a family of potential wells to study the equation under consideration, respectively. Liu et al. [30] introduced a family of potential wells that were different from those in Refs. [27,28,29]. In contrast, the definition of the potential well in this paper differs from those in the above studies.

This paper is organized as follows. Section 1 serves as an introduction. Section 2 provides a comprehensive overview of the relevant functional spaces. Additionally, we define a potential well and give its properties. Section 3 is devoted to the proof of the existence of global solutions by using the theory of potential wells and the Galerkin approximations. In Section 4, we prove the asymptotic behavior of global solutions by using the theory of potential wells and the perturbed energy method [31,32,33]. In Section 5, we summarize our main results.

2. Preliminaries

As in Refs. [34,35], we denote by X the linear space of Lebesgue measurable functions from ${\mathbb{R}}^N$ to $\mathbb{R}$ such that the restriction to $\mathsf{\Omega }$ of any function u in X belongs to $L^2\left(\mathsf{\Omega }\right)$ and

$$\int {\int }_Q\frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+2m}}\mathrm{d}x\mathrm{d}y<\infty ,$$

where $Q:={\mathbb{R}}^{2N}\backslash (\mathcal{C}\mathsf{\Omega }\times \mathcal{C}\mathsf{\Omega })$ and $\mathcal{C}\mathsf{\Omega }:={\mathbb{R}}^N\backslash \mathsf{\Omega }$. The space X is endowed with

$${\parallel u\parallel }_X:={\parallel u\parallel }_{L^2\left(\mathsf{\Omega }\right)}+{\left({\int }_Q\frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+2m}}\mathrm{d}x\mathrm{d}y\right)}^{\frac{1}{2}}.$$

It is straightforward to verify that ${\parallel ·\parallel }_X$ is a norm on X. We introduce the following closed linear subspace of X,

$$X_0:=\{u\in X|u=0\mathrm{a}.\mathrm{e}.\mathrm{in}\mathcal{C}\mathsf{\Omega }\}.$$

This is a Hilbert space equipped with the inner product

$${(u,v)}_{X_0}:=\int {\int }_Q\frac{\left(u\right(x)-u(y\left)\right)\left(v\right(x)-v(y\left)\right)}{{|x-y|}^{N+2m}}\mathrm{d}x\mathrm{d}y$$

and the norm

$${\parallel u\parallel }_{X_0}:=\left(\int {\int }_Q\frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+2m}}\mathrm{d}x\mathrm{d}y\right)\frac{1}{2}$$

which is equivalent to ${\parallel u\parallel }_X$. Moreover, the embedding $X_0\hookrightarrow L^q\left(\mathsf{\Omega }\right)$ is continuous for any $1\le q\le 2_m^{*}$ and compact for any $1\le q<2_m^{*}$, where

$$\begin{array}{c}\hfill 2_m^{*}=\left\{\begin{array}{cc}{\displaystyle \frac{2N}{N-2m}}\hfill & \mathrm{if}2m<N,\hfill \\ \infty \hfill & \mathrm{if}2m\ge N.\hfill \end{array}\right.\end{array}$$

In this paper, the exponents q and r satisfy the following assumption:

$$\left({\mathrm{A}}_1\right)2<r<q<2_m^{*}.$$

In addition, as in Ref. [33], the memory kernel g satisfies

$$\begin{array}{cc}\hfill & \left({\mathrm{A}}_2\right)g\in C^1\left({\mathbb{R}}^{+}\right)\cap L^1\left({\mathbb{R}}^{+}\right),g\left(t\right)\ge 0,g^{\prime }\left(t\right)\le 0\mathrm{for}\mathrm{all}t\in [0,\infty ),\mathrm{and}\hfill \\ \hfill & \kappa :=a-{\int }_0^{\infty }g\left(t\right)\mathrm{d}t>0.\hfill \end{array}$$

For simplicity, we denote

$${\parallel ·\parallel }_p:={\parallel ·\parallel }_{L^p\left(\mathsf{\Omega }\right)},(u,v):={\int }_{\mathsf{\Omega }}uv\mathrm{d}x$$

and

$$(g\circ u)\left(t\right):={\int }_0^{t}g(t-\tau ){\parallel u\left(t\right)-u\left(\tau \right)\parallel }_{X_0}^2\mathrm{d}\tau .$$

Moreover, $C>0$ denotes a generic constant.

Definition 1.

A function $u\in L^{\infty }(0,T;X_0)$ with $u_t\in L^{\infty }(0,T;L^2\left(\mathsf{\Omega }\right))$ is called the weak solution of the problem (1)–(3), if $u\left(0\right)=u_0$ in $X_0$, $u_t\left(0\right)=u_1$ in $L^2\left(\mathsf{\Omega }\right)$, and

$$\begin{array}{cc}\hfill & (u_t\left(t\right),w)+{\int }_0^t{(a+b\parallel u\left(\tau \right)\parallel }_{X_0}^{2p-2}){(u\left(\tau \right),w)}_{X_0}\mathrm{d}\tau -{\int }_0^t{\int }_0^{s}g(s-\tau ){(u\left(\tau \right),w)}_{X_0}\mathrm{d}\tau \mathrm{d}s\hfill \\ \hfill & +(u\left(t\right),w)=(u_1,w)+(u_0,w)+{\int }_0^t{\left(\right|u|}^{q-2}u,w)\mathrm{d}\tau -{\int }_0^t{\left(\right|u|}^{r-2}u,w)\mathrm{d}\tau \hfill \end{array}$$

for any $w\in X_0$ and $t\in (0,T]$.

The energy function for the problem (1)–(3) is defined as follows:

$$\begin{array}{cc}\hfill E\left(t\right):=& \frac{1}{2}\parallel u_t{\left(t\right)\parallel }_2^2+\frac{b}{2p}{\parallel u\left(t\right)\parallel }_{X_0}^{2p}+\frac{1}{2}\left(a-{\int }_0^{t}g\left(\tau \right)\mathrm{d}\tau \right){\parallel u\left(t\right)\parallel }_{X_0}^2\hfill \\ \hfill & +\frac{1}{2}(g\circ u)\left(t\right)-\frac{1}{q}{\parallel u\left(t\right)\parallel }_q^q+\frac{1}{r}{\parallel u\left(t\right)\parallel }_r^r.\hfill \end{array}$$

Following the idea from Ref. [36], the potential well and its boundary are defined by

$$\mathcal{W}:=\left\{u\in X_0\left|{\parallel u\parallel }_{X_0}<{\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{1}{2}}\right.\right\}$$

(5)

and

$$𝜕\mathcal{W}:=\left\{u\in X_0\left|{\parallel u\parallel }_{X_0}={\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{1}{2}}\right.\right\},$$

(6)

where the depth of the potential well is

$$d:=\frac{q-2}{2q}{\kappa }^{\frac{q}{q-2}}{\mathfrak{C}}_1^{-\frac{2q}{q-2}},$$

(7)

and ${\mathfrak{C}}_1$ is the best Sobolev constant for the embedding $X_0\hookrightarrow L^q\left(\mathsf{\Omega }\right)$, i.e.,

$${\mathfrak{C}}_1:=\underset{u\in X_0\backslash \left\{0\right\}}{sup}\frac{{\parallel u\parallel }_q}{{\parallel u\parallel }_{X_0}}.$$

Lemma 1.

Let $\left({\mathrm{A}}_1\right)$ and $\left({\mathrm{A}}_2\right)$ be satisfied. Then,

(i) 

if $u\in \mathcal{W}$ and ${\parallel u\parallel }_{X_0}\ne 0$, then ${\kappa \parallel u\parallel }_{X_0}^2>{\parallel u\parallel }_q^q$;

(ii) 

if $u\in 𝜕\mathcal{W}$, then ${\kappa \parallel u\parallel }_{X_0}^2\ge {\parallel u\parallel }_q^q$.

Proof. 

(i) By $u\in \mathcal{W}$ and (5), we obtain

$${\parallel u\parallel }_{X_0}<{\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{1}{2}}.$$

From (7), we obtain

$${\parallel u\parallel }_{X_0}<{\kappa }^{\frac{1}{q-2}}{\mathfrak{C}}_1^{-\frac{q}{q-2}}.$$

Noting that ${\parallel u\parallel }_{X_0}\ne 0$, we have

$${\kappa \parallel u\parallel }_{X_0}^2>{\mathfrak{C}}_1^q{\parallel u\parallel }_{X_0}^q.$$

Therefore,

$${\kappa \parallel u\parallel }_{X_0}^2>{\parallel u\parallel }_q^q.$$

(ii) By $u\in 𝜕\mathcal{W}$ and (6), we obtain

$${\parallel u\parallel }_{X_0}={\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{1}{2}}.$$

By adopting similar arguments in the proof of (i), it is evident that

$${\kappa \parallel u\parallel }_{X_0}^2\ge {\parallel u\parallel }_q^q.$$

□

3. Existence of Global Solutions

Theorem 1.

Let $\left({\mathrm{A}}_1\right)$ and $\left({\mathrm{A}}_2\right)$ be satisfied. If $u_0\in \mathcal{W}$, $u_1\in L^2\left(\mathsf{\Omega }\right)$, and $E\left(0\right)<d$, then the problem (1)–(3) has a global solution $u\left(t\right)\in \overline{\mathcal{W}}:=\mathcal{W}\cup 𝜕\mathcal{W}$ for all $t\in (0,\infty )$.

Proof. 

We divide the proof of this theorem into three steps.

Step I. Galerkin approximations. Let ${\left\{{\omega }_j\right\}}_{j=1}^{\infty }$ be the orthogonal basis of $X_0$ and an orthonormal basis of $L^2\left(\mathsf{\Omega }\right)$ given by eigenfunctions of ${(-\mathsf{\Delta })}^m$ with boundary condition (3) (see Ref. [34], Proposition 9, for details). Denote $W_n=\mathrm{Span}\{{\omega }_1,{\omega }_2,\cdots ,{\omega }_n\}$, $n=1,2,\cdots$. We seek the approximate solutions of the problem (1)–(3)

$$u_n\left(t\right)=\sum _{j=1}^n{\xi }_{jn}\left(t\right){\omega }_j,n=1,2,\cdots ,$$

(8)

which satisfy

$$\begin{array}{cc}\hfill & (u_{ntt}\left(t\right),w)+(a+b\parallel u_n\left(t\right){\parallel }_{X_0}^{2p-2}){(u_n\left(t\right),w)}_{X_0}-{\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right),w)}_{X_0}\mathrm{d}\tau \hfill \\ \hfill & +(u_{nt}\left(t\right),w)=\left(\right|u_n{\left(t\right)|}^{q-2}{u}_n\left(t\right),w)-(|u_n\left(t\right){|}^{r-2}{u}_n\left(t\right),w),t>0,\hfill \end{array}$$

(9)

$$u_n\left(0\right)=\sum _{j=1}^n{\xi }_{jn}\left(0\right){\omega }_j\to u_0\mathrm{in}{X}_0,$$

(10)

$$u_{nt}\left(0\right)=\sum _{j=1}^n{\xi }_{jn}^{\prime }\left(0\right){\omega }_j\to u_1\mathrm{in}{L}^2\left(\mathsf{\Omega }\right),$$

(11)

for any $w\in W_n$. Let ${\xi }_n\left(t\right)={({\xi }_{1n}\left(t\right),{\xi }_{2n}\left(t\right),\cdots ,{\xi }_{nn}\left(t\right))}^T$. Then, the vector function ${\xi }_n$ solves

$${\xi }_n^{\prime \prime }\left(t\right)+{\xi }_n^{\prime }\left(t\right)+{\mathcal{L}}_n(t,{\xi }_n\left(t\right))=0,t>0,$$

(12)

$${\xi }_n\left(0\right)={((u_0,{\omega }_1),(u_0,{\omega }_2),\cdots ,(u_0,{\omega }_n))}^T,$$

(13)

$${\xi }_n^{\prime }\left(0\right)={((u_1,{\omega }_1),(u_1,{\omega }_2),\cdots ,(u_1,{\omega }_n))}^T,$$

(14)

where

$${\mathcal{L}}_n(t,{\xi }_n\left(t\right))={({\mathcal{L}}_{1n}(t,{\xi }_n\left(t\right)),{\mathcal{L}}_{2n}(t,{\xi }_n\left(t\right)),\cdots ,{\mathcal{L}}_{nn}(t,{\xi }_n\left(t\right)))}^T,$$

$$\begin{array}{cc}\hfill {\mathcal{L}}_{in}(t,{\xi }_n\left(t\right))=& \left(a+b{∥\sum _{j=1}^n{\xi }_{jn}\left(t\right){\omega }_j∥}_{X_0}^{2p-2}\right){\left(\sum _{j=1}^n{\xi }_{jn}\left(t\right){\omega }_j,{\omega }_i\right)}_{X_0}\hfill \\ \hfill & -{\int }_0^{t}g(t-\tau ){\left(\sum _{j=1}^n{\xi }_{jn}\left(\tau \right){\omega }_j,{\omega }_i\right)}_{X_0}\mathrm{d}\tau \hfill \\ \hfill & +\left({\left|\sum _{j=1}^n{\xi }_{jn}\left(t\right){\omega }_j\right|}^{q-2}\sum _{j=1}^n{\xi }_{jn}\left(t\right){\omega }_j,{\omega }_i\right)\hfill \\ \hfill & -\left({\left|\sum _{j=1}^n{\xi }_{jn}\left(t\right){\omega }_j\right|}^{r-2}\sum _{j=1}^n{\xi }_{jn}\left(t\right){\omega }_j,{\omega }_i\right).\hfill \end{array}$$

In light of the standard theory of ordinary differential equations, the problem (12)–(14) has a solution ${\xi }_n\in C^2[0,T_n)$, with $T_n\le T$. Consequently, $u_n\left(t\right)$ defined by (8) satisfies the problem (9)–(11).

Step II. A priori estimates. Taking $w=u_{nt}\left(t\right)$ in (9), we obtain

$$\begin{array}{cc}\hfill & \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\parallel u_{nt}{\left(t\right)\parallel }_2^2+\frac{a}{2}\parallel u_n{\left(t\right)\parallel }_{X_0}^2+\frac{b}{2p}{\parallel u_n\left(t\right)\parallel }_{X_0}^{2p}\right)\hfill \\ \hfill & -{\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right),u_{nt}\left(t\right))}_{X_0}\mathrm{d}\tau +{\parallel u_{nt}\left(t\right)\parallel }_2^2\hfill \\ \hfill & =\frac{1}{q}\frac{\mathrm{d}}{\mathrm{d}t}\parallel u_n{\left(t\right)\parallel }_q^q-\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}t}{\parallel u_n\left(t\right)\parallel }_r^r.\hfill \end{array}$$

(15)

Concerning the fourth term on the left hand side of (15), we have

$$\begin{array}{cc}\hfill & {\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right),u_{nt}\left(t\right))}_{X_0}\mathrm{d}\tau \hfill \\ \hfill =& {\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right)-u_n\left(t\right),u_{nt}\left(t\right))}_{X_0}\mathrm{d}\tau +{\int }_0^{t}g(t-\tau ){(u_n\left(t\right),u_{nt}\left(t\right))}_{X_0}\mathrm{d}\tau \hfill \\ \hfill =& -\frac{1}{2}{\int }_0^{t}g(t-\tau )\frac{\mathrm{d}}{\mathrm{d}t}\parallel u_n\left(\tau \right)-u_n{\left(t\right)\parallel }_{X_0}^2\mathrm{d}\tau +\frac{1}{2}{\int }_0^{t}g(t-\tau )\frac{\mathrm{d}}{\mathrm{d}t}{\parallel u_n\left(t\right)\parallel }_{X_0}^2\mathrm{d}\tau \hfill \\ \hfill =& -\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left((g\circ u_n)\left(t\right)-{\int }_0^{t}g\left(\tau \right)\mathrm{d}\tau {\parallel u_n\left(t\right)\parallel }_{X_0}^2\right)+\frac{1}{2}(g^{\prime }\circ u_n)\left(t\right)-\frac{1}{2}g\left(t\right){\parallel u_n\left(t\right)\parallel }_{X_0}^2.\hfill \end{array}$$

Substituting this equality into (15) and performing integration with respect to t, we obtain

$$E_n\left(t\right)+{\int }_0^t\left(\parallel u_{n\tau }{\left(\tau \right)\parallel }_2^2-\frac{1}{2}(g^{\prime }\circ u_n)\left(\tau \right)+\frac{1}{2}g\left(\tau \right){\parallel u_n\left(\tau \right)\parallel }_{X_0}^2\right)\mathrm{d}\tau =E_n\left(0\right)$$

(16)

for all $t\in [0,T]$, where

$$\begin{array}{cc}\hfill E_n\left(t\right)=& \frac{1}{2}\parallel u_{nt}{\left(t\right)\parallel }_2^2+\frac{b}{2p}\parallel u_n{\left(t\right)\parallel }_{X_0}^{2p}+\frac{1}{2}\left(a-{\int }_0^{t}g\left(\tau \right)\mathrm{d}\tau \right){\parallel u_n\left(t\right)\parallel }_{X_0}^2\hfill \\ \hfill & +\frac{1}{2}(g\circ u_n)\left(t\right)-\frac{1}{q}\parallel u_n\left(t\right){\parallel }_q^q+\frac{1}{r}{\parallel u_n\left(t\right)\parallel }_r^r.\hfill \end{array}$$

(17)

Based on the observations from (10) and (11), we conclude that $E_n\left(0\right)<d$ and $u_n\left(0\right)\in \mathcal{W}$ for a sufficiently large n. We claim that

$$u_n\left(t\right)\in \mathcal{W}$$

(18)

for all $t\in [0,T]$ and a sufficiently large n. Suppose that $u_n\left(t\right)\notin \mathcal{W}$ for some $0<t<T$. Then, there exists a time $0<t_0<T$ such that $u_n\left(t_0\right)\in 𝜕\mathcal{W}$ and $u_n\left(t\right)\in \mathcal{W}$ for all $t\in [0,t_0)$. Therefore,

$$\parallel u_n\left(t_0\right){\parallel }_{X_0}={\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{1}{2}}.$$

Utilizing (17) in conjunction with part (ii) in Lemma 1, we are able to derive

$$\begin{array}{cc}\hfill E_n\left(t_0\right)\ge & \frac{1}{2}\kappa \parallel u_n\left(t_0\right){\parallel }_{X_0}^2-\frac{1}{q}{\parallel u_n\left(t_0\right)\parallel }_q^q\hfill \\ \hfill =& \frac{q-2}{2q}\kappa {\parallel u_n\left(t_0\right)\parallel }_{X_0}^2+\frac{1}{q}\left(\kappa \parallel u_n\left(t_0\right){\parallel }_{X_0}^2-{\parallel u_n\left(t_0\right)\parallel }_q^q\right)\hfill \\ \hfill \ge & \frac{q-2}{2q}\kappa {\parallel u_n\left(t_0\right)\parallel }_{X_0}^2\hfill \\ \hfill =& d.\hfill \end{array}$$

In view of (16), we obtain a contradiction with $E_n\left(0\right)<d$.

By (17), assertion (18) and (i) in Lemma 1, we can conclude that

$$\begin{array}{cc}\hfill E_n\left(t\right)\ge & \frac{1}{2}\parallel u_{nt}{\left(t\right)\parallel }_2^2+\frac{1}{2}\kappa \parallel u_n{\left(t\right)\parallel }_{X_0}^2-\frac{1}{q}{\parallel u_n\left(t\right)\parallel }_q^q\hfill \\ \hfill =& \frac{1}{2}\parallel u_{nt}{\left(t\right)\parallel }_2^2+\frac{q-2}{2q}\kappa {\parallel u_n\left(t\right)\parallel }_{X_0}^2+\frac{1}{q}\left(\kappa \parallel u_n{\left(t\right)\parallel }_{X_0}^2-{\parallel u_n\left(t\right)\parallel }_q^q\right)\hfill \\ \hfill \ge & \frac{1}{2}\parallel u_{nt}{\left(t\right)\parallel }_2^2+\frac{q-2}{2q}\kappa {\parallel u_n\left(t\right)\parallel }_{X_0}^2,\hfill \end{array}$$

(19)

which, together with (16), gives

$$\frac{1}{2}\parallel u_{nt}{\left(t\right)\parallel }_2^2+\frac{q-2}{2q}\kappa {\parallel u_n\left(t\right)\parallel }_{X_0}^2<d$$

for all $t\in [0,T]$. Thus,

$$\parallel u_{nt}{\left(t\right)\parallel }_2^2<2d$$

(20)

and

$$\parallel u_n{\left(t\right)\parallel }_{X_0}^2<\frac{2q}{(q-2)\kappa }d.$$

(21)

Further, it can be derived from (21) that

$$\parallel |u_n{\left(t\right)|}^{q-2}{u}_n{\left(t\right)\parallel }_{\frac{q}{q-1}}=\parallel u_n{\left(t\right)\parallel }_q^{q-1}\le C{\parallel u_n\left(t\right)\parallel }_{X_0}^{q-1}<C{\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{q-1}{2}}$$

(22)

and

$$\parallel |u_n{\left(t\right)|}^{r-2}{u}_n{\left(t\right)\parallel }_{\frac{r}{r-1}}=\parallel u_n{\left(t\right)\parallel }_r^{r-1}\le C{\parallel u_n\left(t\right)\parallel }_{X_0}^{r-1}<C{\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{r-1}{2}}$$

(23)

for all $t\in [0,T]$. Estimates (20)–(23) imply that

$$\left\{u_n\right\}\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;X_0),$$

(24)

$$\left\{u_{nt}\right\}\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^2\left(\mathsf{\Omega }\right)),$$

(25)

$$\left\{\right|u_n{|}^{q-2}{u}_n\}\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^{\frac{q}{q-1}}\left(\mathsf{\Omega }\right)),$$

(26)

$$\left\{\right|u_n{|}^{r-2}{u}_n\}\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^{\frac{r}{r-1}}\left(\mathsf{\Omega }\right)).$$

(27)

Step III. Passage to the limit. By virtue of (24)–(27), there is a subsequence of $\left\{u_n\right\}$ (still represented by $\left\{u_n\right\}$) and a function u such that as $n\to \infty$,

$$u_n\rightharpoonup u\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;X_0),$$

(28)

$$u_{nt}\rightharpoonup u_t\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;L^2\left(\mathsf{\Omega }\right)),$$

(29)

$$|u_n{|}^{q-2}{u}_n\rightharpoonup {\left|u\right|}^{q-2}u\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;L^{\frac{q}{q-1}}\left(\mathsf{\Omega }\right)),$$

$$|u_n{|}^{r-2}{u}_n\rightharpoonup {\left|u\right|}^{r-2}u\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;L^{\frac{r}{r-1}}\left(\mathsf{\Omega }\right)).$$

Integrating (9) with respect to t, we obtain

$$\begin{array}{cc}\hfill & (u_{nt}\left(t\right),w)+{\int }_0^t(a+b\parallel u_n\left(\tau \right){\parallel }_{X_0}^{2p-2}){(u_n\left(\tau \right),w)}_{X_0}\mathrm{d}\tau \hfill \\ \hfill & -{\int }_0^t{\int }_0^{s}g(s-\tau ){(u_n\left(\tau \right),w)}_{X_0}\mathrm{d}\tau \mathrm{d}s+(u_n\left(t\right),w)\hfill \\ \hfill & =(u_{nt}\left(0\right),w)+(u_n\left(0\right),w)+{\int }_0^t\left(\right|u_n\left(\tau \right){|}^{q-2}{u}_n\left(\tau \right),w)\mathrm{d}\tau \hfill \\ \hfill & -{\int }_0^t\left(\right|u_n\left(\tau \right){|}^{r-2}{u}_n\left(\tau \right),w)\mathrm{d}\tau .\hfill \end{array}$$

Taking $n\to \infty$, we obtain

$$\begin{array}{cc}\hfill & (u_t\left(t\right),w)+{\int }_0^t{(a+b\parallel u\left(\tau \right)\parallel }_{X_0}^{2p-2}){(u\left(\tau \right),w)}_{X_0}\mathrm{d}\tau -{\int }_0^t{\int }_0^{s}g(s-\tau ){(u\left(\tau \right),w)}_{X_0}\mathrm{d}\tau \mathrm{d}s\hfill \\ \hfill & +(u\left(t\right),w)=(u_1,w)+(u_0,w)+{\int }_0^t\left(\right|u\left(\tau \right){|}^{q-2}u\left(\tau \right),w)\mathrm{d}\tau -{\int }_0^t\left(\right|u\left(\tau \right){|}^{r-2}u\left(\tau \right),w)\mathrm{d}\tau .\hfill \end{array}$$

From (10) and (11), we obtain $u\left(0\right)=u_0$ in $X_0$ and $u_t\left(0\right)=u_1$ in $L^2\left(\mathsf{\Omega }\right)$. Hence, u is a global solution of the problem (1)–(3). Furthermore, from (28), we obtain

$${\parallel u\left(t\right)\parallel }_{X_0}\le \underset{n\to \infty }{lim\; inf}{\parallel u_n\left(t\right)\parallel }_{X_0},$$

which, together with (21), gives

$${\parallel u\left(t\right)\parallel }_{X_0}\le {\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{1}{2}},$$

that is, $u\left(t\right)\in \overline{\mathcal{W}}$ for all $t\in (0,\infty )$. □

4. Asymptotic Behavior of Global Solutions

Theorem 2.

In addition to all the assumptions of Theorem 1, assume that there exists a constant $\rho >0$ such that $g^{\prime }\left(t\right)\le -\rho g\left(t\right)$ for all $t\in [0,\infty )$. Then,

$${\parallel u\left(t\right)\parallel }_{X_0}^2+{\parallel u_t\left(t\right)\parallel }_2^2\le \alpha e^{-\beta t},\forall t\in [0,\infty ),$$

(30)

for some constants $\alpha ,\beta >0$.

Proof. 

For the approximate solutions given in the proof of Theorem 1, we construct

$$L\left(t\right)=E_n\left(t\right)+\epsilon \mathsf{\Psi }\left(t\right),\forall t\in [0,\infty ),$$

(31)

where $\mathsf{\Psi }\left(t\right)=(u_n\left(t\right),u_{nt}\left(t\right))$, and $\epsilon >0$ is a constant to be determined later.

Step I. We now claim that there exist two constants ${\gamma }_1,{\gamma }_2>0$, depending on $\epsilon$, such that

$${\gamma }_1{E}_n\left(t\right)\le L\left(t\right)\le {\gamma }_2{E}_n\left(t\right),\forall t\in [0,\infty ).$$

(32)

In fact, according to Cauchy’s inequality, we obtain

$$\left|\mathsf{\Psi }\left(t\right)\right|\le \frac{1}{2}\parallel u_n{\left(t\right)\parallel }_2^2+\frac{1}{2}{\parallel u_{nt}\left(t\right)\parallel }_2^2,$$

and so,

$$\left|\mathsf{\Psi }\left(t\right)\right|\le \frac{{\mathfrak{C}}_2^2}{2}\parallel u_n{\left(t\right)\parallel }_{X_0}^2+\frac{1}{2}{\parallel u_{nt}\left(t\right)\parallel }_2^2,$$

(33)

where ${\mathfrak{C}}_2$ is the best Sobolev constant for the embedding $X_0\hookrightarrow L^2\left(\mathsf{\Omega }\right)$. Combining (33) and (19), we obtain $\left|\mathsf{\Psi }\left(t\right)\right|\le M{E}_n\left(t\right)$ for some constant $M>0$ independent of n, which, together with (31), yields that assertion (32) holds.

Step II. We next claim that

$$L^{\prime }\left(t\right)\le -\epsilon \eta E_n\left(t\right)$$

(34)

for sufficiently small $\eta$ and $\epsilon$. Indeed, since

$$E_n^{\prime }\left(t\right)=\frac{1}{2}(g^{\prime }\circ u_n)\left(t\right)-\frac{1}{2}g\left(t\right)\parallel u_n\left(t\right){\parallel }_{X_0}^2-{\parallel u_{nt}\left(t\right)\parallel }_2^2$$

and

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}^{\prime }\left(t\right)=& \parallel u_{nt}{\left(t\right)\parallel }_2^2-b\parallel u_n{\left(t\right)\parallel }_{X_0}^{2p}-a{\parallel u_n\left(t\right)\parallel }_{X_0}^2+{\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right),u_n\left(t\right))}_{X_0}\mathrm{d}\tau \hfill \\ \hfill & -(u_n\left(t\right),u_{nt}\left(t\right))+\parallel u_n{\left(t\right)\parallel }_q^q-{\parallel u_n\left(t\right)\parallel }_r^r,\hfill \end{array}$$

it follows from (31) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L^{\prime }\left(t\right)=& \frac{1}{2}(g^{\prime }\circ u_n)\left(t\right)-\frac{1}{2}g\left(t\right)\parallel u_n\left(t\right){\parallel }_{X_0}^2-\parallel u_{nt}\left(t\right){\parallel }_2^2+\epsilon {\parallel u_{nt}\left(t\right)\parallel }_2^2\hfill \\ \hfill & -\epsilon b\parallel u_n{\left(t\right)\parallel }_{X_0}^{2p}-\epsilon a{\parallel u_n\left(t\right)\parallel }_{X_0}^2+\epsilon {\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right),u_n\left(t\right))}_{X_0}\mathrm{d}\tau \hfill \\ \hfill & -\epsilon (u_n\left(t\right),u_{nt}\left(t\right))+\epsilon \parallel u_n{\left(t\right)\parallel }_q^q-\epsilon {\parallel u_n\left(t\right)\parallel }_r^r.\hfill \end{array}\end{array}$$

(35)

For the seventh term on the right hand side of (35), we can derive from Schwarz’s inequality and Cauchy’s inequality with ${ϵ}_1>0$ that

$$\begin{array}{cc}\hfill & {\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right),u_n\left(t\right))}_{X_0}\mathrm{d}\tau \hfill \\ \hfill =& {\int }_0^{t}g(t-\tau ){\parallel u_n\left(t\right)\parallel }_{X_0}^2\mathrm{d}\tau +{\int }_0^{t}g(t-\tau ){(u_n\left(\tau \right)-u_n\left(t\right),u_n\left(t\right))}_{X_0}\mathrm{d}\tau \hfill \\ \hfill \le & {\int }_0^{t}g\left(\tau \right)\mathrm{d}\tau \parallel u_n{\left(t\right)\parallel }_{X_0}^2+{ϵ}_1{\int }_0^{t}g\left(\tau \right)\mathrm{d}\tau {\parallel u_n\left(t\right)\parallel }_{X_0}^2+\frac{1}{4{ϵ}_1}(g\circ u_n)\left(t\right)\hfill \\ \hfill \le & (a-\kappa )\parallel u_n{\left(t\right)\parallel }_{X_0}^2+{ϵ}_1(a-\kappa ){\parallel u_n\left(t\right)\parallel }_{X_0}^2+\frac{1}{4{ϵ}_1}(g\circ u_n)\left(t\right).\hfill \end{array}$$

(36)

For the eighth term on the right-hand side of (35), we deduce from Cauchy’s inequality with ${ϵ}_2>0$ that

$$\begin{array}{cc}\hfill -(u_n\left(t\right),u_{nt}\left(t\right))\le & {ϵ}_2\parallel u_n{\left(t\right)\parallel }_2^2+\frac{1}{4{ϵ}_2}{\parallel u_{nt}\left(t\right)\parallel }_2^2\hfill \\ \hfill \le & {ϵ}_2{\mathfrak{C}}_2^2\parallel u_n{\left(t\right)\parallel }_{X_0}^2+\frac{1}{4{ϵ}_2}{\parallel u_{nt}\left(t\right)\parallel }_2^2.\hfill \end{array}$$

(37)

Plugging (36) and (37) into (35), we obtain

$$\begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & \left(\epsilon +\frac{\epsilon }{4{ϵ}_2}-1\right)\parallel u_{nt}{\left(t\right)\parallel }_2^2-\epsilon b{\parallel u_n\left(t\right)\parallel }_{X_0}^{2p}\hfill \\ \hfill & +\epsilon ({ϵ}_1(a-\kappa )+{ϵ}_2{\mathfrak{C}}_2^2-\kappa ){\parallel u_n\left(t\right)\parallel }_{X_0}^2+\left(\frac{\epsilon }{4{ϵ}_1}-\frac{\rho }{2}\right)(g\circ u_n)\left(t\right)\hfill \\ \hfill & +\epsilon \parallel u_n{\left(t\right)\parallel }_q^q-\epsilon {\parallel u_n\left(t\right)\parallel }_r^r,\hfill \end{array}$$

and so,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & -\epsilon \eta E_n\left(t\right)+\left(\epsilon +\frac{\epsilon }{4{ϵ}_2}+\frac{\epsilon \eta }{2}-1\right)\parallel u_{nt}{\left(t\right)\parallel }_2^2+\epsilon b\left(\frac{\eta }{2p}-1\right){\parallel u_n\left(t\right)\parallel }_{X_0}^{2p}\hfill \\ \hfill & +\epsilon \left({ϵ}_1(a-\kappa )+{ϵ}_2{\mathfrak{C}}_2^2+\frac{\eta a}{2}-\kappa \right){\parallel u_n\left(t\right)\parallel }_{X_0}^2\hfill \\ \hfill & +\left(\frac{\epsilon }{4{ϵ}_1}+\frac{\epsilon \eta }{2}-\frac{\rho }{2}\right)(g\circ u_n)\left(t\right)+\epsilon \parallel u_n\left(t\right){\parallel }_q^q-\epsilon {\parallel u_n\left(t\right)\parallel }_r^r\hfill \\ \hfill & -\frac{\epsilon \eta }{q}\parallel u_n{\left(t\right)\parallel }_q^q+\frac{\epsilon \eta }{r}{\parallel u_n\left(t\right)\parallel }_r^r,\hfill \end{array}\end{array}$$

(38)

where $\eta >0$ is a constant to be determined later. We conclude from (16) and (19) that

$$E_n\left(0\right)\ge \frac{q-2}{2q}\kappa {\parallel u_n\left(t\right)\parallel }_{X_0}^2.$$

This leads to

$$\parallel u_n{\left(t\right)\parallel }_{X_0}\le {\left(\frac{2q}{(q-2)\kappa }{E}_n\left(0\right)\right)}^{\frac{1}{2}}.$$

Hence,

$$\parallel u_n{\left(t\right)\parallel }_q^q\le {\mathfrak{C}}_1^q\parallel u_n{\left(t\right)\parallel }_{X_0}^{q-2}\parallel u_n{\left(t\right)\parallel }_{X_0}^2\le {\mathfrak{C}}_1^q{\left(\frac{2q}{(q-2)\kappa }{E}_n\left(0\right)\right)}^{\frac{q-2}{2}}{\parallel u_n\left(t\right)\parallel }_{X_0}^2.$$

The substitution of this inequality into (38) yields

$$\begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & -\epsilon \eta E_n\left(t\right)+\left(\epsilon +\frac{\epsilon }{4{ϵ}_2}+\frac{\epsilon \eta }{2}-1\right)\parallel u_{nt}{\left(t\right)\parallel }_2^2+\epsilon b\left(\frac{\eta }{2p}-1\right){\parallel u_n\left(t\right)\parallel }_{X_0}^{2p}\hfill \\ \hfill & +\epsilon \left({ϵ}_1(a-\kappa )+{ϵ}_2{\mathfrak{C}}_2^2+\frac{\eta a}{2}+{\mathfrak{C}}_1^q{\left(\frac{2q}{(q-2)\kappa }{E}_n\left(0\right)\right)}^{\frac{q-2}{2}}-\kappa \right){\parallel u_n\left(t\right)\parallel }_{X_0}^2\hfill \\ \hfill & +\left(\frac{\epsilon }{4{ϵ}_1}+\frac{\epsilon \eta }{2}-\frac{\rho }{2}\right)(g\circ u_n)\left(t\right)-\epsilon \parallel u_n\left(t\right){\parallel }_r^r+\frac{\epsilon \eta }{r}{\parallel u_n\left(t\right)\parallel }_r^r.\hfill \end{array}$$

(39)

Since

$${\mathfrak{C}}_1^q{\left(\frac{2q}{(q-2)\kappa }{E}_n\left(0\right)\right)}^{\frac{q-2}{2}}<{\mathfrak{C}}_1^q{\left(\frac{2q}{(q-2)\kappa }d\right)}^{\frac{q-2}{2}}=\kappa ,$$

the values of ${ϵ}_1$, ${ϵ}_2$, and $\eta$ can be selected to be sufficiently small, such that $\eta <min\{2p,r\}$ and

$${ϵ}_1(a-\kappa )+{ϵ}_2{\mathfrak{C}}_2^2+\frac{\eta a}{2}+{\mathfrak{C}}_1^q{\left(\frac{2q}{(q-2)\kappa }{E}_n\left(0\right)\right)}^{\frac{q-2}{2}}-\kappa <0.$$

Consequently, it follows from (39) that

$$\begin{array}{cc}\hfill L^{\prime }\left(t\right)\le & -\epsilon \eta E_n\left(t\right)+\left(\epsilon +\frac{\epsilon }{4{ϵ}_2}+\frac{\epsilon \eta }{2}-1\right){\parallel u_{nt}\left(t\right)\parallel }_2^2\hfill \\ \hfill & +\left(\frac{\epsilon }{4{ϵ}_1}+\frac{\epsilon \eta }{2}-\frac{\rho }{2}\right)(g\circ u_n)\left(t\right).\hfill \end{array}$$

Thus, for fixed ${ϵ}_1$, ${ϵ}_2$, and $\eta$, we can choose

$$\epsilon <min\left\{\frac{1}{M},\frac{4{ϵ}_2}{4{ϵ}_2+1+2\eta {ϵ}_2},\frac{2\rho {ϵ}_1}{1+2\eta {ϵ}_1}\right\}$$

such that assertion (34) holds.

Step III. We prove (30). To accomplish this, by combining (34) with the second inequality stated in assertion (32), we obtain

$$L^{\prime }\left(t\right)\le -\frac{\epsilon \eta }{{\gamma }_2}L\left(t\right).$$

Hence,

$$L\left(t\right)\le C{e}^{-\frac{\epsilon \eta }{{\gamma }_2}t},\forall t\in [0,\infty ).$$

The first inequality in assertion (32) allows us to further deduce that

$$E_n\left(t\right)\le \frac{C}{{\gamma }_1}{e}^{-\frac{\epsilon \eta }{{\gamma }_2}t},\forall t\in [0,\infty ).$$

(40)

It can be inferred from (28) and (29) that

$${\parallel u\left(t\right)\parallel }_{X_0}^2+{\parallel u_t\left(t\right)\parallel }_2^2\le \underset{n\to \infty }{lim\; inf}\left(\parallel u_n{\left(t\right)\parallel }_{X_0}^2+{\parallel u_{nt}\left(t\right)\parallel }_2^2\right),$$

which, together with (19) and (40), gives

$$\underset{n\to \infty }{lim\; inf}\left(\parallel u_n{\left(t\right)\parallel }_{X_0}^2+{\parallel u_{nt}\left(t\right)\parallel }_2^2\right)\le \underset{n\to \infty }{lim\; inf}C{E}_n\left(t\right)\le \frac{C}{{\gamma }_1}{e}^{-\frac{\epsilon \eta }{{\gamma }_2}t},$$

and so, (30) is derived, where $\alpha =C/{\gamma }_1$ and $\beta =\epsilon \eta /{\gamma }_2$. Theorem 2 is proved. □

Theorem 2 provides a sufficient condition for the asymptotic behavior of solutions of the problem (1)–(3). For example, we take $N=1$, $m=1/2$, $a=1$, $b=0$, $q=4$, $r=3$, and $g\left(t\right)=g\left(0\right)e^{-t}$ with $g\left(0\right)<1$. Then, it is easy to verify that $\left({\mathrm{A}}_1\right)$, $\left({\mathrm{A}}_2\right)$, and $g^{\prime }\left(t\right)\le -\rho g\left(t\right)$ hold, where $\kappa =1-g\left(0\right)$ and $\rho =1$. By virtue of (7) and (5), we obtain

$$d=\frac{1}{4}{(1-g\left(0\right))}^2{\mathfrak{C}}_1^{-4}$$

and

$$\mathcal{W}=\left\{u\in X_0\left|{\parallel u\parallel }_{X_0}<{(1-g\left(0\right))}^{\frac{1}{2}}{\mathfrak{C}}_1^{-2}\right.\right\}.$$

Therefore, if $u_0$ and $u_1$ satisfy

$$\parallel u_0{\parallel }_{X_0}<{(1-g\left(0\right))}^{\frac{1}{2}}{\mathfrak{C}}_1^{-2}$$

and

$$\frac{1}{2}\parallel u_1{\parallel }_2^2+\frac{1}{2}\parallel u_0{\parallel }_{X_0}^2-\frac{1}{4}\parallel u_0{\parallel }_4^4+\frac{1}{3}{\parallel u_0\parallel }_3^3<\frac{1}{4}{(1-g\left(0\right))}^2{\mathfrak{C}}_1^{-4},$$

then, we conclude from Theorem 2 that the norm of global solutions of the problem (1)–(3) in the phase space $X_0\times L^2\left(\mathsf{\Omega }\right)$ decays exponentially to zero when the time tends to infinity.

5. Conclusions

The initial boundary value problem for a class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs is studied by this paper. In the case where the two opposing external forces ${\left|u\right|}^{q-2}u$ and $-{\left|u\right|}^{r-2}u$ appear simultaneously, the existence and asymptotic behavior of global solutions are derived, namely, Theorems 1 and 2. More specifically, Theorem 1 shows that if the initial data $u_0$ lie in the potential well, and the initial energy is less than the depth of the potential well, then the problem (1)–(3) has a global solution. Theorem 2 shows us that if the memory kernel decays exponentially, then the global solutions of the problem (1)–(3) also do this.

Our main technical tool is the theory of potential wells, which is different from the classical. We describe the potential well by a sphere, whose radius is expressed by d. Although the depth of the potential well d is smaller than the classical, the spatial structure of the potential well is clearer so that it is not necessary to introduce the Nehari functional and the Nehari manifold.

In the future, we will focus on the study of the qualitative properties of solutions of fractional viscoelastic Kirchhoff equations involving more general Kirchhoff functions and nonlinear source terms.