A Few Remarks About a Theorem by J. Rauch

Abstract

In this paper, we consider semi-linear hyperbolic initial boundary value problem on multidimensional domains. We assume that the system is symmetric hyperbolic, with maximal dissipative boundary conditions, the boundary is either characteristic of constant multiplicity or noncharacteristic. In particular, we treat the case of conservative boundary conditions. We show that this problem can be seen as a limit when e → 0 + of a parabolic initial boundary value problem. The parabolic operators are obtained from the hyperbolic operator by adding a viscosity eE, where E is a well chosen elliptic second order operator. We prescribe a Dirichlet boundary condition for these parabolic perturbations. This answers a question raised by J. Rauch in [13]. The elliptic operators E verify a weakly dissipation assumption. On characteristics components, strict dissipation is required. We also give a topological description of the set of the convenient symmetric viscosities for vacuum Maxwell's system with incoming wave condition.