[Problème de Cauchy global pour des systèmes hyperboliques á coefficients superlinéaire lorsque ]
Nous étudions les problèmes de Cauchy bien posès pour les systèmes hyperboliques linéaires du 1er ordre avec des racines caractéristiques superlinéaires lorsque . 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 , 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 . 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 and provide explicit examples proving that our estimates are sharp.
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@article{CRMATH_2009__347_1-2_49_0,
author = {Gourdin, Daniel and Gramchev, Todor},
title = {Global {Cauchy} problems for hyperbolic systems with characteristics admitting superlinear growth for $ |x|\to \infty $},
journal = {Comptes Rendus. Math\'ematique},
pages = {49--54},
publisher = {Elsevier},
volume = {347},
number = {1-2},
year = {2009},
doi = {10.1016/j.crma.2008.11.009},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2008.11.009/}
}
TY - JOUR AU - Gourdin, Daniel AU - Gramchev, Todor TI - Global Cauchy problems for hyperbolic systems with characteristics admitting superlinear growth for $ |x|\to \infty $ JO - Comptes Rendus. Mathématique PY - 2009 SP - 49 EP - 54 VL - 347 IS - 1-2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2008.11.009/ DO - 10.1016/j.crma.2008.11.009 LA - en ID - CRMATH_2009__347_1-2_49_0 ER -
%0 Journal Article %A Gourdin, Daniel %A Gramchev, Todor %T Global Cauchy problems for hyperbolic systems with characteristics admitting superlinear growth for $ |x|\to \infty $ %J Comptes Rendus. Mathématique %D 2009 %P 49-54 %V 347 %N 1-2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2008.11.009/ %R 10.1016/j.crma.2008.11.009 %G en %F CRMATH_2009__347_1-2_49_0
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. http://archive.numdam.org/articles/10.1016/j.crma.2008.11.009/
[1] Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam–New York, 1982
[2] The Technique of Pseudodifferential Operators, LMS Lecture Note Series, vol. 202, Cambridge Univ. Press, Cambridge, 1995
[3] Cauchy problem for SG-hyperbolic equations with constant multiplicities, Ricerche Mat., Volume 48 (1999) no. suppl., pp. 25-43
[4] Problème de Cauchy non caractéristique pour les systèmes hyperboliques à caractéristiques de multiplicité variable, Domaine de dépendence, Comm. Partial Differential Equations, Volume 4 (1979), pp. 447-507
[5] Global in time solutions of evolution equations in scales of Banach function spaces in , Bull. Sci. Math. (2007)
[6] Solutions globales d'un problème de Cauchy linèaire, J. Funct. Anal., Volume 202 (2003) no. 1, pp. 123-146
[7] Ordinary Differential Equations, John Wiley & Sons, New York–London–Sydney, 1964
[8] Representation of solutions and regularity properties for weakly hyperbolic systems, Pseudo-Differential Operators and Related Topics, Oper. Theory Adv. Appl., vol. 164, Birkhäuser, Basel, 2006, pp. 53-63
[9] On the Cauchy Problem, Notes and Reports in Mathematics in Science and Engineering, vol. 3, Science Press and Academic Press, 1985
[10] Global -boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations, Volume 31 (2006), pp. 547-569
[11] The nonlocal existence problem of ordinary differential equations, Amer. J. Math., Volume 67 (1945), pp. 277-284 (125–132)
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