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

Abstract

In this paper, we consider a doubly nonlinear parabolic equation ∂tβ(u) − ∇ • α(x, ∇u) ∋ f with the homogeneous Dirichlet boundary condition in a bounded domain, where β : R → 2 R is a maximal monotone graph satisfying 0 ∈ β(0) and ∇•α(x, ∇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 β. 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 ∈ (1, ∞) without any conditions for β except 0 ∈ β(0). We also discuss the uniqueness of solution by using properties of entropy solution.