The Cauchy problem for weakly parabolic systems (Q1380081)

The purpose of this paper is to investigate the notion of weak parabolicity of first order, nonlinear systems of general type, and to give sufficient conditions for the local solvability to the initial value problem with \(C^\infty\), or real analytic, initial data. We first find a suitable Levi type condition on the lower order terms, fulfilled by several hyperbolic-parabolic systems, which ensures the local well-posedness in the Sobolev classes. Then we prove a result of solvability in the analytic class, without Levi conditions, for a special class of semilinear, weakly parabolic systems. Finally, we give some stability results with respect to small perturbations of the initial data.