We study a stationary thermistor mo
del
describing the electrothermal behavior of organic semiconductor
devices featuring non-Ohmic current–voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the
device is mo
deled by an Arrhenius-like temperature
depen
dency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is mo
deled by a power law such that the electrical conductivity
depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a
p(x)den">de">-Laplace-type problem with piecewise constant exponent.
We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the p(x)den">de">-Laplacian. Finally, Schauder’s fixed-point theorem is used to show the existence of solutions.