Scattering and self-similar solutions for the nonlinear wave equation

Abstract

We study scattering theory and self-similar solutions for the nonlinear wave equation $\Box u+\lambda|u|^{p-1}u=0$ in two or three space dimensions under the assumption $p_0(n) <p <1+4/(n-1)$, where $p_0(n)$ is the larger root of the equation $(n-1)p^2-(n+1)p-2=0$. The relation between the theory of scattering and that of self-similar solutions is considered from the point of view of asymptotically free solutions and asymptotically self-similar solutions.