Systems Describing Electrothermal Effects with $p(x)$-Laplacian-like Structure for Discontinuous Variable Exponents

We 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 systems are 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 simultaneously overcome two obstacles---the discontinuous variable exponent and the L¹ right-hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.