Uniqueness of entropy solution for general anisotropic convection-diffusion problems

This work is an attempt to develop the uniqueness theory of entropy solution for the Cauchy problem associated to a general non-isotropic nonlinear strongly degenerate parabolic-hyperbolic equation. Our aim is to extend, at the same time, results of [1] and [11]. The novelty in this paper is the fact that we are dealing with general anisotropic diffusion problems, not necessarily with Lipschitz convection-diffusion flux functions in the whole space. Moreover, the source term depends on the unknown function of the problem. Under an abstract lemma and an additional assumption, we ensure the comparison principle which leads us to the uniqueness. In unbounded domains without Lipschitz condition on the convection and diffusion flux functions, this assumption seems to be optimal to establish uniqueness (cf. [1], [3], [14]).