In the last decade, the study of double phase problems (involving (p,q) - Laplace) has attracted great interest due to their ability to model complex phenomena and materials with heterogeneous properties, e.g. composites made of two different materials. From a mathematical point of view, double phase elliptic and parabolic equations, even for the constant exponents p and q (isotropic case), attract great interest due to their so-called nonstandard growth conditions, for which most of the well-established methods of nonlinear analysis are no longer applicable. In the anisotropic and variable exponents settings, the situation becomes even more complex. This creates a strong motivation to develop new analytical tools tailored to these equations. Numerous publications in the last years highlight the active development of double phase problems, see [Boegelein, Duzaar, Marcellini, Scheven, JMPA, 2022; Boegelein, Strunk, Annali di Mat. Pura ed Appl., 2024; Crespo-Blanco, Gasinski, Harjulehto, Winkert, JDE, 2022; De Filippis, G. Mingione, ARMA, 2023; Buryachenko, Skrypnik, Potential An., 2022, J. Math.Sci., 2017].

In this talk we study a coupled double phase (p(x), q(x)) - Laplace thermistor model, that describes the electrothermal behavior of semiconductor devices, taking different parallel charge transport mechanisms into account. Such systems model materials conducting both heat and electrical current and for which the electrical conductivity in the definition of current density can strongly depend on the temperature. Devices of this type are called thermistors, see, for instance, [Cimatti, Quant. Appl. Math., 1989]. As for coupled systems involving double phase equations, and, in particular, thermistor type models, such problems are addressed in the present study for the first time.

More precisely, our system consists of the current flow elliptic double phase equation, involving (p(x), q(x)) Laplacian with modulation coefficient a(x) and conductivity coefficients depend on the temperature as well as the phases of the materials, for the electrostatic potential u, coupled to the heat equation for the temperature T. The right hand side of the heat equation, Joule heat term H, depends on u and T and belongs to L¹ . The system is complemented by Dirichlet and homogeneous Neumann boundary conditions for the electrical potential, as well as Robin boundary condition for the temperature.

We operate with discontinuous (only measurable) variable exponents p(x), q(x) and modulation coefficient a(x), and investigate the full (p(x), q(x)) double phase situation for a(x)>0. Together with the necessity to involve the concept of entropy solutions to the heat equation with right hand side from the space L¹, that are the main challenges of the problem.

Our main result concerns the existence of a weak solution (u,T) to the coupled double phase (p(x), q(x)) - Laplace thermistor model. The solution u, electrostatic potential, is assumed as a weak solution (in a generalized Sobolev space, related to a Musielak-Orlicz space) to the double phase elliptic equation with variable measurable exponents p(x), q(x), modulation a(x)>0 and conductivity coefficients. The solution T is an entropy solution to the heat equation with nonlinear right hand side H from the space L¹ . To tackle the problem, we use a combination of entropy solution techniques and Schauder's fixed-point theorem. We discuss also the electrothermal behavior of heterogeneous organic semiconductor materials with different charge transport mechanisms characterized by different activation energies. Note, that p(x)- Laplace thermistor models were studied very well before [Bulicek, Glitzky, Liero, SIAM J. Math. Anal, 2016; DCDS-S, 2017; Glitzky, Liero, Nonlin. Anal., 2017; DCDS-S, 2021; Liero, Koprucki, Fischer, Scholz, Glitzky, ZAMP, 2015].

Acknowledgment. This work is supported by the Alexander von Humboldt‑Stiftung.