Sharp existence results for self-similar solutions of semilinear wave equations

Abstract

The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation \(u_{tt} - \Delta u = f(u)\) with \(f(u) = \gamma |u|^{\alpha+1}\) or \(f(u) = \gamma|u|^{\alpha}u\) in R 3×R + for small Cauchy data is proven if \(\sqrt{2} < \alpha < 2\). A counterexample is given which shows that the lower bound on α is sharp.