Partial Differential Equations
Global Cauchy problems for hyperbolic systems with characteristics admitting superlinear growth for |x|→∞
[Problème de Cauchy global pour des systèmes hyperboliques á coefficients superlinéaire lorsque |x|→+∞]

Présenté par : Yvonne Choquet-Bruhat

Gourdin, Daniel1 ; Gramchev, Todor2
1 Département de mathématiques, Université de Paris 6, 175, rue Chevaleret, 75013 Paris, France
2 Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Comptes Rendus. Mathématique, Tome 347 (2009) no. 1-2, pp. 49-54.

Nous étudions les problèmes de Cauchy bien posès pour les systèmes hyperboliques linéaires du 1er ordre avec des racines caractéristiques superlinéaires lorsque |x|→∞. On introduit des hypothèses sur la croissance superlinéaire inspirée des théorèmes de A. Wintner en 1945 pour les solutions globales d'équations différentielles ordinaires et nous montrons que le résultat est acquis pour deux types d'espaces de Sobolev avec poids. Nous construisons une transformation du type de Liouville globale des variables d'espace qui réduit le problème au cas de coefficients bornés. En consequence nous montrons la propagation a vitesse finie et l'existence d'un domaine fini de dépendance. On montre le blow up des solutions près de t=0, avec des exemples explicites montrant que les estimations sont pointues.

We investigate the global well-posedness of the Cauchy problem for first order linear hyperbolic systems allowing superlinear growth of the characteristic roots for |x|→+∞. We introduce hypotheses on the superlinear growth inspired by theorems of A. Wintner in 1945 for global solutions of ODEs and show global well-posedness of the Cauchy problem in two types of new weighted Sobolev spaces. We construct a change of the space variables of a global Liouville type which reduces to the case of bounded coefficients. As an outcome, we derive finite propagation speed and the existence of finite domains of dependence for hyperbolic systems of differential equations. We exhibit also instant blow-up of solutions near t=0 and provide explicit examples proving that our estimates are sharp.

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DOI : 10.1016/j.crma.2008.11.009
Gourdin, Daniel 1 ; Gramchev, Todor 2

1 Département de mathématiques, Université de Paris 6, 175, rue Chevaleret, 75013 Paris, France
2 Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy 
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Gourdin, Daniel; Gramchev, Todor. Global Cauchy problems for hyperbolic systems with characteristics admitting superlinear growth for $ |x|\to \infty $. Comptes Rendus. Mathématique, Tome 347 (2009) no. 1-2, pp. 49-54. doi : 10.1016/j.crma.2008.11.009. https://www.numdam.org/articles/10.1016/j.crma.2008.11.009/

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