Well-posedness results for triply nonlinear degenerate parabolic equations

Abstract

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 = u 0 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 ) a 0 ( ∇ ϕ ( u ) ) , with a nonlinearity ϕ which is strictly increasing except on a locally finite number of segments, and the nonlinearity a 0 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 u 0 , 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 u 0 , 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.