Blow-Up of Positive Solutions to Wave Equations in High Space Dimensions

TL;DR

This paper investigates the conditions under which positive solutions to semilinear wave equations in high dimensions blow up, extending previous results and addressing technical challenges in deriving lower bounds for solutions with non-zero initial data.

Contribution

It generalizes and extends prior blow-up results for wave equations to higher dimensions with non-zero initial data, especially in even dimensions.

Findings

Positive solutions blow up under certain conditions in high dimensions.

Extended blow-up results to cases with non-zero initial position and velocity.

Addressed technical challenges in deriving lower bounds for solutions in even dimensions.

Abstract

This paper is concerned with the Cauchy problem for the semilinear wave equation: $u_{tt}-\Delta u=F(u) \ \mbox{in} \ R^n\times[0, \infty)$, where the space dimension $n \ge 2$, $F(u)=|u|^p$ 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(1995) and Takamura, Uesaka and Wakasa(2011). 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.