Entropy Solutions for Nonlinear Degenerate Elliptic-parabolic-hyperbolic Problems

We consider the nonlinear degenerate elliptic-parabolic-hyperbolic equation ∂tg(u)−∆b(u)− div Φ(u) = f(g(u)) in (0, T )× Ω, where g and b are nondecreasing continuous functions, Φ is vectorial and continuous, and f is Lipschitz continuous. We prove the existence, comparison and uniqueness of entropy solutions for the associated initial-boundary-value problem where Ω is a bounded domain in RN . For the associated initial-value problem where Ω = RN , N ≥ 3, the existence of entropy solutions is proved. Moreover, for the case when Φ ◦ g−1 is locally Hölder continuous of order 1 − 1/N , and |b(u)| ≤ ω(|g(u)|), where ω is nondecreasing continuous with ω(0) = 0, we can prove the L1-contraction principle, and hence the uniqueness.