Asymptotic behavior of solutions of some nonlinearly damped wave equations on $\mathbb R^N$

Abstract

We discuss the asymptotic behavior of solutions of the nonlinearly damped wave equation $$ u_{tt} +\delta \vert u_t\vert ^{m-1}u_t -\phi (x)\Delta u = \lambda u\vert u\vert ^{\beta -1}, \quad x \in \mathbb R^n, \ t \geq 0, $$ with the initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3$, $ \delta > 0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{p}(\mathbb R^n)\cap L^{\infty}(\mathbb R^n)$, for some $p$. We prove blow-up of solutions when the source term dominates over the damping, and the initial energy is assumed to be positive. We also discuss global existence energy decay of solutions.