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 -δu = F(u) in ℝn × [0,∞); where the space dimension n ≥2, F(u) = |u|^p or F(u) = |u|^p-1u 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.