Entropy formulation of degenerate parabolic equation with zero-flux boundary condition

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We consider the general degenerate hyperbolic-parabolic equation: $$u_t + {\rm div} f(u) - \Delta \phi(u) = 0\; {\rm in} Q = (0, T) \times \Omega, \quad T > 0, \quad \Omega \subset \mathbb{R}^N;$$ with initial condition and the zero flux boundary condition. Here \({\phi}\) is a continuous non-decreasing function. Following Burger et al. (J Math Anal Appl 326:108–120, 2007), we assume that f is compactly supported (this is the case in several applications), and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension N ≥ 1 under a partial genuine nonlinearity assumption on f. Uniqueness is shown for the case N = 1, using the idea of Andreianov and Bouhsiss (J Evol Equ 4:273–295, 2004), nonlinear semigroup theory and a specific regularity result for one dimension.