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

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.