Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Thermistor systems of $p(x)$-Laplace-type with discontinuous exponents via entropy solutions

Pages: 697 - 713, Volume 10, Issue 4, August 2017      doi:10.3934/dcdss.2017035

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Miroslav Bulíček - Mathematical Institute, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic (email)
Annegret Glitzky - Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany (email)
Matthias Liero - Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany (email)

Abstract: We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of $p(x)$-Laplacian-type with discontinuous exponent $p$, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a~priori $L^1$ term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

Keywords:  Sobolev spaces with variable exponent, existence of weak solution, entropy solution, thermistor system, $p(x)$-Laplacian, heat transfer.
Mathematics Subject Classification:  35J92, 35Q79, 80A20.

Received: April 2016;      Revised: September 2016;      Available Online: April 2017.