Asymptotics and analytic modes for the wave equation in similarity coordinates

TL;DR

This paper analyzes the long-term behavior of solutions to the radial wave equation in similarity coordinates, demonstrating that analytic eigenfunctions dominate the dynamics and establishing the linear stability of a fundamental self-similar solution.

Contribution

It provides a rigorous analysis of the spectral properties of the wave equation in similarity coordinates and proves the linear stability of the self-similar solution with precise decay rates.

Findings

Long time behavior is governed by analytic eigenfunctions.

Established linear stability of the self-similar solution.

Derived sharp decay rates for perturbations.

Abstract

We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues with analytic eigenfunctions. Our results show that, for sufficiently regular data, the long time behaviour of the solution is governed by the analytic eigenfunctions. The same techniques are applied to the linear stability problem for the fundamental self--similar solution $\chi_T$ of the wave equation with a focusing power nonlinearity. Analogous to the free wave equation, we show that the long time behaviour (in similarity coordinates) of linear perturbations around $\chi_T$ is governed by analytic mode solutions. In particular, this yields a rigorous proof for the linear stability of $\chi_T$ with the sharp decay rate for the perturbations.