Abstract: We study the Cauchy problem in R^N for the parabolic equation u_t+div F(u)=Δφ(u), which can degenerate into a hyperbolic equation for some intervals of values of u.

In the context of conservation laws (the case φ≡0), it is known that an entropy solution can be non-unique when F' has singularities. We show the uniqueness of an entropy solution to the general parabolic problem for all L^∞ initial datum, under the isotropic condition on the flux F known for conservation laws. The only assumption on the diffusion term is that φ is a non-decreasing continuous function.