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

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


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.


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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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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