A few remarks on a theorem by J. Rauch Franck Sueur 35L5035B2535K50boundary layersmaximale dissipativeweakly dissipationcharacteristic boundaryviscous perturbations 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 $varepsilon o 0^{+}$ of a parabolic initial boundary value problem. The parabolic operators are obtained from the hyperbolic operator by adding a viscosity $varepsilonmathcal{E}$, where $mathcal{E}$ 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 cite{13}. The elliptic operators $mathcal{E}$ 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. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2666 10.1512/iumj.2005.54.2666 en Indiana Univ. Math. J. 54 (2005) 1107 - 1144 state-of-the-art mathematics http://iumj.org/access/