Scattering for systems of semilinear wave equations with different speeds of propagation

Abstract

We give an extension of the a priori estimate, obtained in [8], for a solution of the inhomogeneous wave equation in ${\bf R}^n\times{\bf R}$, where $n=2$ or $n=3$. As an application, we study the asymptotic behavior as $t \to \pm \infty$ of solutions to systems of semilinear wave equations. The discrepancy of the speeds of propagation may make a significant difference from the case of common propagation speeds. (See also Theorem 3.3 and 3.4). Whether such a phenomenon occurs or not depends on the type of the interaction determined by the nonlinearities.