Blow-up phenomena for a class of extensible beam equations

Abstract

In this paper, we investigate the initial boundary value problem of the following nonlinear extensible beam equation with nonlinear damping term $$u_{t t}+\Delta^2 u-M\left(\|\nabla u\|^2\right) \Delta u-\Delta u_t+\left|u_t\right|^{r-1} u_t=|u|^{p-1} u$$ which was considered by Yang et al. (Advanced Nonlinear Studies 2022; 22:436-468). We consider the problem with the nonlinear damping and establish the finite time blow-up of the solution for the initial data at arbitrary high energy level, including the estimate lower and upper bounds of the blowup time. The result provides some affirmative answer to the open problems given in (Advanced Nonlinear Studies 2022; 22:436-468).