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 form $\left(d/dt\right)𝒜\left(u\right)+ℬ\left(u\right)\ni f\left(t\right)$ in $V^{\prime }$, $t\in \left(0, T\right]$, where $V$ is a real reflexive Banach space, $𝒜$ and $ℬ$ are maximal monotone operators (possibly multivalued) from $V$ to its dual $V^{\prime }$. In view of some practical applications, we assume that $𝒜$ and $ℬ$ are subdifferentials. By using the back difference approximation, existence is established, and our proof relies on the continuity of $𝒜$ and the coerciveness of $ℬ$. As an application, we give the existence for a nonlinear degenerate parabolic equation.