Obstacle problems for a coupling of quasilinear hyperbolic-parabolic equations

set in a parabolic area Qp = ]0, T [ × (Ω \ Ωh), complementary to the former, and for suitable conditions across the interface between the two regions Qp and Qh. The geometrical configuration is such that:Ω = Ωh∪Ωp;Ωh andΩp are two disjoint bounded domains with Lipschitz boundaries denoted Γl , for l in {h, p}. In addition, the interface Γhp = Γh ∩ Γp is Lipschitz and such that Hn−1(Γ hp ∩ Γl \ Γhp) = 0, where for q in [0, n + 1], H denotes the q-dimensional Hausdorff measure. For a given threshold θ , the (bilateral) obstacle problem for Th and Tp may be formally written in the free boundary formulation: find a bounded measurable function u on Q ≡ ]0, T [ ×Ω such