Blow Up of Classical Solutions to $\Box$ U=|u|1+α in Three Space Dimensions

Abstract

We study the life span of classical solutions to ◻u = |u|^{1+α} in three space dimensions with initial data t = 0: u = εf(x), u, = εg(x), where f and g have compact support and are not both identically zero, ε is a small parameter. We obtain respectively upper and lower bounds of the same order of magnitude for the life span for sufficiently small ε in case 1 ≤ α ≤ \sqrt{2}. We also proved that the classical solution always blows up even when ε = 1 in the critical case α = \sqrt{2}.