Scattering for semilinear wave equations with small data in three space dimensions
Solutions of a class of semilinear wave equations in three space dimensions with power-type nonlinearity are shown to have the same asymptotic behaviour as suitable solutions of the corresponding linear problem as \(t\to \pm \infty\). The main assumptions are that the Cauchy data are small and smooth enough and that the nonlinearity vanishes of order \(p>1+\sqrt{2}\) at the origin. Thus the scattering operator is shown to exist in a dense set in the neighbourhood of the origin in energy space. The result is sharp in the sense that blow-up occurs in the case of the nonlinearity \(| u| ^ p\) if \(p<1+\sqrt{2}\) as was shown by F. John.
scientific article; zbMATH DE number 4019610