We study well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problems of the kind
b(u)t−div𝔞̃ (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 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,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 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.

degenerate elliptic-hyperbolic-parabolic equation; Leray-Lions type operator; homogeneous Dirichlet problem; entropy solution; well-posedness; continuous dependence on data