Asymptotics and analytic modes for the wave equation in similarity coordinates

Autor(en)

Roland Donninger

Abstrakt

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.

Organisation(en)

Gravitationsphysik

Journal

Journal of Evolution Equations

Band

9

Seiten

511-523

Anzahl der Seiten

13

ISSN

1424-3199

DOI

https://doi.org/10.1007/s00028-009-0022-x

Publikationsdatum

2009

Peer-reviewed

Ja

ÖFOS 2012

1010 Mathematik, 1030 Physik, Astronomie

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/adbf5c4d-73c7-4797-97e8-f890d8c33169