On mild and weak solutions of elliptic-parabolic problems

We consider an elliptic-parabolic equation in divergence form $b(v)_t = \text{div}\alpha(v, Dv)+ f$ with Dirichlet boundary conditions and initial condition. Under rather general assumptions, we prove existence of mild solutions satisfying an $L^1$-comparison principle; under some additional conditions, these solutions are shown to be weak solutions. Moreover, under the general assumptions, uniqueness of integral solutions is established; under certain conditions, we show that weak solutions are integral solutions. The notions of mild and integral solutions are derived from nonlinear semigroup theory; by this approach, we extend and make precise former results on existence and uniqueness of weak solutions.