Mathematical Analysis of some Hyperbolic-Parabolic Inner Obstacle Problems

We establish the existence and uniqueness of the solution to some inner obstacle problems for a coupling of a multidimensional quasilinear firstorder hyperbolic equation set in a region Ωh with a quasilinear parabolic one set in the complementary Ωp = Ω\Ωh. We start by providing the definition of a weak solution through an entropy inequality on the whole domain. Since the interface ∂Ωp ∩ ∂Ωh contains the outward characteristics for the first-order operator in Ωh, the uniqueness proof begins by considering first the hyperbolic zone and then the parabolic one. The existence property uses the vanishing viscosity method and to pass to the limit on the hyperbolic zone, we refer to the notion of process solution.