The cauchy problem for nonlinear hyperbolic equations with Levi condition

✍ Scribed by Massimo Cicognani; Luisa Zanghirati

• Publisher: Elsevier Science
• Year: 1999
• Tongue: French
• Weight: 142 KB
• Volume: 123
• Category: Article
• ISSN: 0007-4497
• DOI: 10.1016/s0007-4497(99)00114-1

✦ Synopsis

Manuscript presented by M.P. MALLIAVIN, received in Januar 1998 ABSTRACT. -The Cauchy problem for weakly hyperbolic equations is generally not C ∞ well posed without assuming conditions on lower order terms: this is well known since the famous E.E. Levi paper [7], generalized many years later by several authors.

Here we want to study the same problem in nonlinear framework, hence it is natural to impose "Levi conditions" on the linearized operator. We shall confine ourselves to consider equations with constant multiplicity for which Levi conditions are plain (see for example J. Chazarian [2], H. Flascka and G. Strang [3], S. Mizohata and Y. Ohya [9], J. Vaillant [12]) and several applications to Mathematical Physics are possible.

As far as we know, the only result of this type is proved by D. Gourdin [4], where he treats, with different methods, a class of equations having small intersection with the one we consider here. © Elsevier, Paris

1. Main results

We shall study local solvability for the quasilinear Cauchy problem: |α| m a α t, x, D m u D α t,x u = f t, x, D m u , D j t u |t =0 = g j , 0 j < m,