Existence for Nonlinear Evolution Equations and Application to Degenerate Parabolic Equation

Abstract

We consider an abstract Cauchy problem for a doubly nonlinear evolution equation of the formd/dt𝒜u+ℬu∋ftinV′,t∈0, T, whereVis a real reflexive Banach space,𝒜andℬare maximal monotone operators (possibly multivalued) fromVto its dualV′. In view of some practical applications, we assume that𝒜andℬare subdifferentials. By using the back difference approximation, existence is established, and our proof relies on the continuity of𝒜and the coerciveness ofℬ. As an application, we give the existence for a nonlinear degenerate parabolic equation.