Asymptotic bahavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear wave systems with multiple speeds in three dimensions, showing that solutions are asymptotically free in energy but may differ pointwise from free solutions.

Contribution

It demonstrates the asymptotic freeness of solutions in energy norm and characterizes asymptotically free solutions in arbitrary dimensions for such systems.

Findings

Solutions are asymptotically free in the energy sense.

Pointwise behavior of solutions can differ from free solutions.

A general theorem characterizes asymptotically free solutions in any dimension.

Abstract

We consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. Under the null condition for such systems, the global existence of small amplitude solutions is known. In this paper, we will show that the global solution is asymptotically free in the energy sense, by obtaining the asymptotic pointwise behavior of the derivatives of the solution. Nonetheless we can also show that the pointwise behavior of the solution itself may be quite different from that of the free solution. In connection with the above results, a theorem is also developed to characterize asymptotically free solutions for wave equations in arbitrary space dimensions.