Solvability of Doubly Nonlinear Parabolic Equation with $p$-Laplacian

Abstract

In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $\beta : \mathbb{R} \to 2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in \beta (0)$ and $ \nabla \cdot \alpha (x , \nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $\beta $. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for $1 < p < 2$. Main purpose of this paper is to show the solvability of the initial boundary value problem for any $ p \in (1, \infty ) $ without any conditions for $\beta $ except $0 \in \beta (0)$. We also discuss the uniqueness of solution by using properties of entropy solution.