Finite time blow-up for quasilinear wave equations with nonlinear dissipation

Abstract

In this paper we consider a class of quasilinear wave equations \[u_{tt}-\Delta_{\alpha} u-\omega_1\Delta u_t-\omega_2\Delta_{\beta}u_t+\mu\vert u_t\vert^{m-2}u_t=\vert u\vert^{p-2}u,\]associated with initial and Dirichlet boundary conditions. Under certain conditions on \(\alpha,\beta,m,p\), we show that any solution with positive initial energy, blows up in finite time. Furthermore, a lower bound for the blow-up time will be given.