On a nonlocal quasilinear parabolic model related to a current-carrying stellarator
type: Item
instance of: scholarly article
title: On a nonlocal quasilinear parabolic model related to a current-carrying stellarator
zbMATH Open document ID: 1029.35137
DOI: 10.1016/S1468-1218(01)00045-1
author: M. B. Lerena
author: J. I. Díaz
author: Juan Francisco Padial
published in: Nonlinear Analysis. Real World Applications
publication date: 2003-03-11T00:00:00Z
review text: The main result is the existence of the weak solution (in the sense defined in the paper) of the nonlinear parabolic problem \[ \beta(u)_t - \Delta u = G(u)(t,x) + J(u)(t,x) \quad \text{in} \quad (0,T) \times \Omega, \tag{1} \] \[ u(t,x) = \gamma \quad \text{on} \quad (0,T) \times \partial \Omega, \tag{2} \] \[ \beta(u(0,x)) = \beta(u_0(x)) \quad \text{in} \quad \Omega. \tag{3} \] Here \(\Omega \subset \mathbb{R}^2\) is a bounded regular domain, \(T>0\), \(\beta(s) = \min (s,0)\) and \(G(u)\) is defined by means of an integral -- so it is a nonlocal operator. The formulation (1), (2), (3) is related to a problem arising in the study of the magnetic confinement of a plasma in Stellarator device, when the plasma is assumed to be a perfect conductor but with non-zero net current inside each flux magnetic surface. The reviewed paper is closely related to \textit{J. I. Diaz, J. F. Padial} and \textit{J. M. Rakotoson} [Nonlinear Anal., Theory Methods Appl. 34, 857-887 (1998; Zbl 0946.35119)], where the associated stationary problem is studied.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1880132
zbMATH Keywords: quasilinear nonlocal elliptic-parabolic equations
zbMATH Keywords: relative rearrangement
zbMATH Keywords: Galerkin method
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
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full work available at URL: s1468-1218(01)00045-1
OpenAlex ID: W2034587054
description: scientific article; zbMATH DE number 1880132