Solutions of wave equations in the radiation regime

TL;DR

This paper investigates the solutions of wave equations in Minkowski space-time with complex initial data, establishing existence, asymptotic behavior, and polyhomogeneity of solutions at null infinity.

Contribution

It introduces methods to handle singular and polyhomogeneous initial data for wave equations, extending understanding of asymptotic solutions in conformal compactifications.

Findings

Existence of solutions with controlled asymptotic behavior.

Polyhomogeneous initial data lead to polyhomogeneous solutions at null infinity.

Results apply to a broad class of nonlinear hyperbolic systems.

Abstract

We study the ``hyperboloidal Cauchy problem'' for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda\phi^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behaviour, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary Scri of the Minkowski space-time.