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Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

  • H. Moussa
  • F. Ortegón Gallego
  • M. Rhoudaf
Open Access
Article

Abstract

The existence of a capacity solution to a coupled nonlinear parabolic–elliptic system is analyzed, the elliptic part in the parabolic equation being of the form \(-\,\mathrm{div}\, a(x,t,u,\nabla u)\). The growth and the coercivity conditions on the monotone vector field a are prescribed by an N-function, M, which does not have to satisfy a \(\Delta _2\) condition. Therefore we work with Orlicz–Sobolev spaces which are not necessarily reflexive. We use Schauder’s fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.

Keywords

Capacity solution Coupled system Nonlinear parabolic–elliptic equations Weak solutions Orlicz–Sobolev spaces Thermistor problem 

Mathematics Subject Classification

35K60 35D05 35J70 46E30 

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Département de Mathématiques, Faculté des SciencesUniversité Moulay IsmaïlZitoune, MeknesMorocco
  2. 2.Departamento de Matemáticas, Facultad de CienciasUniversidad de CádizPuerto Real, CádizSpain

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