Solvability of Doubly Nonlinear Parabolic Equation with \(p\)-Laplacian

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...

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Bibliographic Details
Published in:arXiv.org 2020-10
Main Author: Uchida, Shun
Format: Article
Language:English
Subjects:
Boundary conditions
Boundary value problems
Coercivity
Dirichlet problem
Entropy of solution
Online Access:Get full text
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Summary: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.
ISSN:2331-8422