Global Cauchy problems for hyperbolic systems with characteristics admitting superlinear growth for [formula omitted

Abstract

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. To cite this article: D. Gourdin, T. Gramchev, C. R. Acad. Sci. Paris, Ser. I 347 (2009).