Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient.
type: Item
instance of: scholarly article
title: Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient.
zbMATH Open document ID: 1042.35092
DOI: 10.1016/S0022-0396(03)00022-6
author: S. Carl
author: Dumitru Motreanu
published in: Journal of Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: Let \(\Omega \subset \mathbb{R}^N\) be a bounded domain with Lipschitz boundary \(\partial \Omega\); \(Q = \Omega \times (0,\tau)\) and \(\Gamma = \partial \Omega \times (0,\tau)\). The authors consider the following initial boundary value problem: \[ \begin{aligned} \frac{\partial u}{\partial t} + Au + \partial g(\cdot,\cdot,u) \ni Fu + h &\text{in }Q, \cr u(\cdot,0) = 0 &\text{in }\Omega \cr u = 0 &\text{on }\Gamma \end{aligned} \] where \(A\) is a second-order quasilinear differential operator in divergence form of Leray-Lions type, \(F\) is a Nemytski operator associated with a Carathéodory function \(f:Q \times \mathbb{R} \rightarrow \mathbb{R}\), and the function \(g:Q \times \mathbb{R} \rightarrow \mathbb{R}\) is locally Lipschitz in the last variable. The notation \(\partial g\) stands for the generalized Clarke gradient w.r.t. the third variable. It is proved that the problem admits extremal solutions within the order interval formed by given lower and upper solutions.
Mathematics Subject Classification ID: 35R70
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K30
Mathematics Subject Classification ID: 35K35
zbMATH DE Number: 1915672
zbMATH Keywords: quasilinear parabolic inclusion
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: extremal solution
zbMATH Keywords: upper solution
zbMATH Keywords: lower solution
zbMATH Keywords: pseudo-monotone operator
zbMATH Keywords: generalized gradient
zbMATH Keywords: comparison
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915672