Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation, with application to the backward-forward stochastic differential utility

We study the deterministic counterpart of a backward-forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type in two spatial variables, with a rank-1 diffusion and a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in general, strong solutions fail to be globally.