Well-posedness results for triply nonlinear degenerate parabolic equations

We study well-posedness of triply nonlinear degenerate elliptic–parabolic–hyperbolic problems of the kindb(u)t−diva˜(u,∇ϕ(u))+ψ(u)=f,u|t=0=u0 in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities b,ϕb,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 a0a0 which is of the Leray–Lions kind. We are interested in existence, uniqueness and stability of L∞L∞ entropy solutions. For the parabolic–hyperbolic equation (b=Idb=Id), we obtain a general continuous dependence result on data u0,fu0,f and nonlinearities b,ψ,ϕ,a˜. Similar result is shown for the degenerate elliptic problem, which corresponds to the case of b≡0b≡0 and general non-decreasing surjective ψ  . Existence, uniqueness and continuous dependence on data u0,fu0,f are shown in more generality. For instance, the assumptions [b+ψ](R)=R[b+ψ](R)=R and the continuity of ϕ○[b+ψ]−1ϕ○[b+ψ]−1 permit to achieve the well-posedness result for bounded entropy solutions of this triply nonlinear evolution problem.