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

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.