Study of a Viscoelastic Wave Equation with a Strong Damping and Variable Exponents

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.

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