Solvability of doubly nonlinear parabolic equation with p-laplacian

Shun Uchida

doi:10.3934/eect.2021033

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2022, Volume 11, Issue 3: 975-1000. Doi: 10.3934/eect.2021033

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Solvability of doubly nonlinear parabolic equation with p-laplacian

Shun Uchida

Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, Japan

Received: October 31, 2020

Published: June 2022

The author is supported by the Fund for the Promotion of Joint International Research (Fostering Joint International Research (B)) #18KK0073, JSPS Japan

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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 studied 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.

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Mathematics Subject Classification: Primary 35K92; Secondary 35K61, 47J35, 34G25.

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