Well-posedness results for triply nonlinear degenerate parabolic equations
Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen and Stanislas Ouaro
Abstract: We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem
b(u)t − div ã(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 ã falls within the Leray–Lions framework. Some restrictions are imposed on the dependence of ã(u,∇φ(u)) on u and also on the set where φ degenerates. A model case is ã(u,∇φ(u))=f̃(b(u),ψ(u),φ(u))+k(u)a0(∇φ(u)), with φ which is strictly increasing except on a locally finite number of segments, and a0 which is of the Leray–Lions kind. We are interested in existence, uniqueness and stability of entropy solutions. If b=Id, we obtain a general continuous dependence result on data u0,f and nonlinearities b,ψ,φ,ã. 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 when [b+ψ](R)=R and φ∘[b+ψ]−1 is continuous.