The goal of the present paper is to study the viscoelastic wave equation with variable exponents

$$\begin{aligned} u_{tt}-\Delta _{p(x)}u-\Delta u+\int _0^tg(t-s)\Delta u(s)\mathrm{{d}}s-\Delta u_t=|u|^{q(x)-2}u \end{aligned}$$

under initial-boundary value conditions, where the exponents of nonlinearity p(x) and q(x) are given functions. To be more precise, blow-up in finite time is proved, upper and lower bounds of the blow-up time are obtained as well. The global existence of weak solutions is presented, moreover, a general stability of solutions is obtained. This work generalizes and improves earlier results in the literature.

中文翻译：

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强阻尼变指数粘弹性波方程的研究

本论文的目的是研究具有变指数的粘弹性波动方程

$$\begin{aligned} u_{tt}-\Delta _{p(x)}u-\Delta u+\int _0^tg(ts)\Delta u(s)\mathrm{{d}}s-\ Delta u_t=|u|^{q(x)-2}u \end{aligned}$$

在初始边界值条件下，非线性指数p ( x ) 和q ( x ) 是给定的函数。更准确地说，证明了有限时间内的爆破，并得到了爆破时间的上下界。给出了弱解的全局存在性，并得到了解的一般稳定性。这项工作概括并改进了文献中的早期结果。