Electron. J. Differential Equations, Vol. 2018 (2018), No. 121, pp. 1-36.

Nonlinear parabolic-elliptic system in Musielak-Orlicz-Sobolev spaces

Francisco Ortegon Gallego, Mohamed Rhoudaf, Hajar Sabiki

The existence of a capacity solution to the thermistor problem in the context of inhomogeneous Musielak-Orlicz-Sobolev spaces is analyzed. This is a coupled parabolic-elliptic system of nonlinear PDEs whose unknowns are the temperature inside a semiconductor material, $u$, and the electric potential, $\varphi$. We study the general case where the nonlinear elliptic operator in the parabolic equation is of the form $Au=-\hbox{div} a(x,t,u,\nabla u)$, A being a Leray-Lions operator defined on $W_0^{1,x}L_M(Q_T)$, where M is a generalized N-function.

Submitted December 26, 2017. Published June 15, 2018.
Math Subject Classifications: 35J70, 35K61, 46E30, 35M13.
Key Words: Parabolic-elliptic system; Musielak-Orlicz-Sobolev spaces; weak solutions; capacity solutions.

An addendum was posted on August 14, 2019. It corrects the proof of an inequality. See the last two pages of this article.

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Francisco Ortegón Gallego
Departamento de Matemáticas
Facultad de Ciencias, Universidad de Cádiz
Campus del Río San Pedro
11510 Puerto Real, Cádiz, Spain
email: francisco.ortegon@uca.es
Mohamed Rhoudaf
Faculté des Sciences de Meknès
Université Moulay-Ismaïl - Meknès
Équipe: EDPs et Calcul Scientifique, Marocco
email: rhoudafmohamed@gmail.com
Hajar Sabiki
Laboratoire d'Analyse
Géométrie et Applications
Faculté des Sciences
BP 133 Kénitra 14000, Marocco
email: sabikihajar@gmail.com

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