Uniqueness of entropy solutions for nonlinear degenerate parabolic problems

Abstract

We consider the general degenerate parabolic equation:

\( u_t - \Delta b(u) + div F(u) = f \quad \mathrm{in} \quad Q \in ]0, T [\times \mathbb{R}^N, T > 0 \)

We prove existence of Kruzkhov entropy solutions of the associated Cauchy problem for bounded data where the flux function F is supposed to be continuous. Uniqueness is established under some additional assumptions on the modulus of continuity of F and b.