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 \[ \partial_t^2u-\Delta u+\frac{\mu}{1+t}\partial_tu=|u|^p \] in $\mathbb{R}_{+}\times\mathbb{R}^n$. More precisely, for $0\leq\mu\leq 2,\mu \neq1$ and $n\geq 2$, 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.