A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation ut=÷ a(u, Du), where a(z,ξ)=∇ξf(z,ξ), and f is a convex function of ξ with linear growth as ||ξ||→∞, satisfying other additional assumptions. In particular, this class includes the case where f(z,ξ)=φ(z)ψ(ξ), φ>0, and ψ is a convex function with linear growth as ||ξ||→∞.