Équations aux dérivées partielles
Existence globale pour une classe d'équations d'ondes perturbées
[Globale existence for a class of semilinear perturbed wave equations]
Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 27-30.

In this paper we prove a global well-posedness result for the following Cauchy problem:

 ${\partial }_{\mathrm{tt}}\mathrm{u}-\Delta \mathrm{u}+{\mathrm{a}}_{0}{\partial }_{t}\mathrm{u}+\sum _{\mathrm{i}=1}^{3}{a}_{i}{\partial }_{{x}_{i}}\mathrm{u}+\mathrm{Vu}=-\mathrm{u}{|\mathrm{u}|}^{\alpha -1},\phantom{\rule{10.0pt}{0ex}}\mathrm{for}\phantom{\rule{3.30002pt}{0ex}}\left(\mathrm{t},\mathrm{x}\right)\in {ℝ}_{t}×{ℝ}_{x}^{3},\phantom{\rule{10.0pt}{0ex}}\mathrm{u}\left(0\right)=\mathrm{f},\phantom{\rule{10.0pt}{0ex}}{\partial }_{t}\mathrm{u}\left(0\right)=\mathrm{g},$
where the initial data $\left(\mathrm{f},\mathrm{g}\right)\in {\stackrel{˙}{H}}^{1}\left({ℝ}^{3}\right)×{\mathrm{L}}^{2}\left({ℝ}^{3}\right)$ are compactly supported, 1⩽α<5, ${a}_{i}\left(\mathrm{t},\mathrm{x}\right)\in {\mathrm{L}}^{\infty }\left({ℝ}_{t}×{ℝ}_{x}^{3}\right)$, $V\left(t,x\right)\in {L}^{\infty }\left({ℝ}_{t},{L}^{3}\left({ℝ}_{x}^{3}\right)\right)$.

Dans cet article nous prouvons que le problème de Cauchy suivant est bien posé :

 ${\partial }_{\mathrm{tt}}\mathrm{u}-\Delta \mathrm{u}+{\mathrm{a}}_{0}{\partial }_{t}\mathrm{u}+\sum _{\mathrm{i}=1}^{3}{a}_{i}{\partial }_{{x}_{i}}\mathrm{u}+\mathrm{Vu}=-\mathrm{u}{|\mathrm{u}|}^{\alpha -1},\phantom{\rule{10.0pt}{0ex}}\mathrm{pour}\phantom{\rule{3.30002pt}{0ex}}\left(\mathrm{t},\mathrm{x}\right)\in {ℝ}_{t}×{ℝ}_{x}^{3},\phantom{\rule{10.0pt}{0ex}}\mathrm{u}\left(0\right)=\mathrm{f},\phantom{\rule{10.0pt}{0ex}}{\partial }_{t}\mathrm{u}\left(0\right)=\mathrm{g},$
$\left(\mathrm{f},\mathrm{g}\right)\in {\stackrel{˙}{H}}^{1}\left({ℝ}^{3}\right)×{\mathrm{L}}^{2}\left({ℝ}^{3}\right)$ sont à support compact, $1⩽\alpha <5,{\mathrm{a}}_{i}\left(\mathrm{t},\mathrm{x}\right)\in {\mathrm{L}}^{\infty }\left({ℝ}_{t}×{ℝ}_{x}^{3}\right),\mathrm{V}\left(\mathrm{t},\mathrm{x}\right)\in {\mathrm{L}}^{\infty }\left({ℝ}_{t},{\mathrm{L}}^{3}\left({ℝ}_{x}^{3}\right)\right)$.

Accepted:
Published online:
DOI: 10.1016/j.crma.2003.11.007
Visciglia, Nicola 1

1 Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 2, 56100 Pisa, Italie
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Visciglia, Nicola. Existence globale pour une classe d'équations d'ondes perturbées. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 27-30. doi : 10.1016/j.crma.2003.11.007. http://www.numdam.org/articles/10.1016/j.crma.2003.11.007/

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