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- ItemSystems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bulíc̆ek, Miroslav; Glitzky, Annegret; Liero, MatthiasWe consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L1 term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles
- ItemThermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bulicek, Miroslav; Glitzky, Annegret; Liero, MatthiasWe 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 L1 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.