2020/10/20 by Shun Uchida, Uchida, Shun · 2 citations
Computer Science · Engineering · Mathematics · #34G25 #47J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary 35K92 #Secondary 35K61 #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2010.10020
openalex publication_date 2020/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider a doubly nonlinear parabolic equation ∂ t β(u) - ∇ ⋅ α(x , ∇ u) \ni 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.