Blow-up of positive solutions to wave equations in high space dimensions

Abstract

This paper is concerned with the Cauchy problem for the semilinear wave equation: $$ u_{tt}-\Delta u=F(u) \quad \mbox{in}\quad \mathbb R^n\times[0,\infty), $$ where the space dimension $n\geq 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p > 1$. Here, the Cauchy data are non-zero and non-compactly supported. Our results on the blow-up of positive radial solutions (not necessarily radial in low dimensions $n=2, 3$) generalize and extend the results of Takamura [19] for zero initial position and Takamura, Uesaka and Wakasa [21] for zero initial velocity. The main technical difficulty in the paper lies in obtaining the lower bounds for the free solution when both initial position and initial velocity are non-identically zero in even space dimensions.