Well-posedness results for triply nonlinear degenerate parabolic equations.
Résumé
We study well-posedness of triply nonlinear degenerate elliptic–parabolic–hyperbolic problems of the kind b(u)t −div a(u,∇φ(u))+ ψ(u) = f , u|t=0 = u0 in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities b,φ and ψ are supposed to be continuous non-decreasing, and the nonlinearity ˜a falls within the Leray–Lions framework. Some restrictions are imposed on the dependence of a(u,∇φ(u)) on u and also on the set where φ degenerates. A model case is a(u,∇φ(u)) = f(b(u),ψ(u),φ(u)) + k(u)a0(∇φ(u)), with a nonlinearity φ which is strictly increasing except on a locally finite number of segments, and the nonlinearity a0 which is of the Leray–Lions kind. We are interested in existence, uniqueness and stability of L∞ entropy solutions. For the parabolic–hyperbolic equation (b = Id), we obtain a general continuous dependence result on data u0, f and nonlinearities b,ψ,φ, a. Similar result is shown for the degenerate elliptic problem, which corresponds to the case of b ≡ 0 and general non-decreasing surjective ψ. Existence, uniqueness and continuous dependence on data u0, f are shown in more generality. For instance, the assumptions [b + ψ](R) = R and the continuity of φ ◦([b + ψ]^{−1}) permit to achieve the well-posedness result for bounded entropy solutions of this triply nonlinear evolution problem.