Blow up of solutions for semilinear wave equations with noneffective damping

TL;DR

This paper investigates conditions under which solutions to a semilinear wave equation with time-dependent damping blow up in finite time, extending previous results and providing improved lifespan estimates for certain damping parameters.

Contribution

It extends prior work by establishing blow-up results for a broader damping range and derives sharper lifespan estimates using novel methods.

Findings

No global solutions for certain p ranges

Lifespan estimates depend on damping parameter μ

Extension of blow-up results to wider damping range

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.