Breathers for nonlinear wave equations

Smiley, Michael W.

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Tome 82 (1988), p. 431-435 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

The semilinear differential equation (1), (2), (3), in $\mathbb{R} \times \Omega$ with $\Omega \in \mathbb{R}^{N}$, (a nonlinear wave equation) is studied. In particular for $\Omega = \mathbb{R}^{3}$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x) = \nu(t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of Cesari.

Résumé

Si studiano equazioni differenziali semilineari (1), (2), (3), in $\mathbb{R} \times \Omega$ con $\Omega \in \mathbb{R}^{N}$ (equazioni delle onde nonlineari). In particolare, per $\Omega = \mathbb{R}^{3}$, si dimostra l'esistenza di soluzioni $u(t,x)$ deboli, periodiche di periodo $T$, non costanti rispetto a $t$, e radiali nelle variabili spaziali, cioè della forma $u(t,x) = U(t,|x|)$. La dimostrazione è basata su una interpretazione distribuzionale di una equazione lineare corrispondente al problema dato, sul criterio di Paley-Wiener per la trasformazione di Laplace, e sul metodo alternativo di Cesari.

Détail

Publié le : 1988-09-01

MR 1151695 | Zbl 0734.35052

Bibliographie

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