Well-posedness Results for Triply Nonlinear Degenerate Parabolic Equations
نویسنده
چکیده
We study the well-posedness of triply nonlinear degenerate ellipticparabolic-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. Date: October 13, 2008. 2000 Mathematics Subject Classification. Primary 35K65; Secondary 35A05.
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