Blow up of solutions for semilinear wave equations with noneffective damping

Abstract

In this paper, we study the finite-time blow up of solutions to the following semilinear wave equation with time-dependent damping \[ \partial_t^2u-Δu+\fracμ{1+t}\partial_tu=|u|^p \] in ${\mathbb{R}}_{+}\times {\mathbb{R}}^n$. More precisely, for $0\le \mu \le 2,\mu \ne 1$ and $n\ge 2$, there is no global solution for $1<p<p_S(n+\mu )$, where $p_S(k)$ is the $k$-dimensional Strauss exponent and a life-span of the blow up solution will be obtained. Our work is an extension of \cite{IS}, where the authors proved a similar blow up result with a larger range of $\mu$. However, we obtain a better life-span estimate when $\mu \in (0,1)\cup (1,2)$ by using a different method.

11 pages