On behavior of solutions to a Petrovsky equation with damping and variable-exponent source

Abstract

This paper deals with the following Petrovsky equation with damping and nonlinear source \[u_{tt}+\Delta^2 u-M(\|\nabla u\|_2^2)\Delta u-\Delta u_t+|u_t|^{m(x)-2}u_t=|u|^{p(x)-2}u\] under initial-boundary value conditions, where $M(s)=a+ bs^\gamma$ is a positive $C^1$ function with parameters $a>0,~b>0,~\gamma\geq 1$, and $m(x),~p(x)$ are given measurable functions. The upper bound of the blow-up time is derived for low initial energy using the differential inequality technique. For $m(x)\equiv2$, in particular, the upper bound of the blow-up time is obtained by the combination of Levine's concavity method and some differential inequalities under high initial energy. In addition, by making full use of the strong damping, the lower bound of the blow-up time is discussed. Moreover, the global existence of solutions and an energy decay estimate are presented by establishing some energy estimates and by exploiting a key integral inequality.