# American Institute of Mathematical Sciences

• Previous Article
Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case
• DCDS-S Home
• This Issue
• Next Article
Volume constrained minimizers of the fractional perimeter with a potential energy
August 2017, 10(4): 697-713. doi: 10.3934/dcdss.2017035

## Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions

 1 Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83,186 75 Praha 8, Czech Republic 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39,10117 Berlin, Germany

* Corresponding author: matthias.liero@wias-berlin.de

Received  April 2016 Revised  September 2016 Published  April 2017

We 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.

Citation: Miroslav Bulíček, Annegret Glitzky, Matthias Liero. Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 697-713. doi: 10.3934/dcdss.2017035
##### References:
 [1] L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551. doi: 10.1016/S0294-1449(16)30113-5. [2] M. Bulíček, A. Glitzky and M. Liero, Systems describing electrothermal effects with p(x)-Laplace like structure for discontinuous variable exponents, SIAM J. Math. Anal., 48 (2016), 3496-3514. doi: 10.1137/16M1062211. [3] G. Cimatti, Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math., 47 (1989), 117-121. doi: 10.1090/qam/987900. [4] G. Cimatti, Remark on the thermistor problem with rapidly growing conductivity, Applicable Analysis, 80 (2007), 133-140. doi: 10.1080/00036810108840985. [5] L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017, Springer, Berlin, 2011. doi: 10.1007/978-3-642-18363-8. [6] A. Fischer, T. Koprucki, K. Gärtner, J. Brückner, B. Lüssem, K. Leo, A. Glitzky and R. Scholz, Feel the heat: Nonlinear electrothermal feedback in organic LEDs, Adv. Funct. Mater., 24 (2014), 3367-3374. doi: 10.1002/adfm.201303066. [7] A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, T. Koprucki, K. Gärtner and A. Glitzky, Self-heating, bistability, and thermal switching in organic semiconductors, Phys. Rev. Lett., 110 (2013), 126601. doi: 10.1103/PhysRevLett.110.126601. [8] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1975. doi: 10.1002/mana.19750672207. [9] A. Glitzky and M. Liero, Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices, Nonlinear Anal. Real World Appl., 34 (2017), 536-562. doi: 10.1016/j.nonrwa.2016.09.015. [10] A. Kufner, O. John and S. Fučik, Function Spaces, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977. [11] C. Leone and A. Porretta, Entropy solutions for nonlinear elliptic equations in L1, Nonlinear Analysis, Theory, Methods & Applications, 32 (1998), 325-334. doi: 10.1016/S0362-546X(96)00323-9. [12] M. Liero, T. Koprucki, A. Fischer, R. Scholz and A. Glitzky, p-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices, Zeitschrift für Angewandte Mathematik und Physik, 66 (2015), 2957-2977. doi: 10.1007/s00033-015-0560-8. [13] P. Lindqvist, Notes on the p-Laplace Equation, Report 102, University of Jyväskyla, Department of Mathematics and Statistics, ISBN: 951-39-2586-2,2006. [14] L. Orsina, Elliptic equations with measure data, 2013, URL http://www1.mat.uniroma1.it/people/orsina/AS1213/AS1213.pdf. [15] T. Roubíček, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics, 153, Springer, Basel, 2013.

show all references

##### References:
 [1] L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551. doi: 10.1016/S0294-1449(16)30113-5. [2] M. Bulíček, A. Glitzky and M. Liero, Systems describing electrothermal effects with p(x)-Laplace like structure for discontinuous variable exponents, SIAM J. Math. Anal., 48 (2016), 3496-3514. doi: 10.1137/16M1062211. [3] G. Cimatti, Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math., 47 (1989), 117-121. doi: 10.1090/qam/987900. [4] G. Cimatti, Remark on the thermistor problem with rapidly growing conductivity, Applicable Analysis, 80 (2007), 133-140. doi: 10.1080/00036810108840985. [5] L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017, Springer, Berlin, 2011. doi: 10.1007/978-3-642-18363-8. [6] A. Fischer, T. Koprucki, K. Gärtner, J. Brückner, B. Lüssem, K. Leo, A. Glitzky and R. Scholz, Feel the heat: Nonlinear electrothermal feedback in organic LEDs, Adv. Funct. Mater., 24 (2014), 3367-3374. doi: 10.1002/adfm.201303066. [7] A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, T. Koprucki, K. Gärtner and A. Glitzky, Self-heating, bistability, and thermal switching in organic semiconductors, Phys. Rev. Lett., 110 (2013), 126601. doi: 10.1103/PhysRevLett.110.126601. [8] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1975. doi: 10.1002/mana.19750672207. [9] A. Glitzky and M. Liero, Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices, Nonlinear Anal. Real World Appl., 34 (2017), 536-562. doi: 10.1016/j.nonrwa.2016.09.015. [10] A. Kufner, O. John and S. Fučik, Function Spaces, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977. [11] C. Leone and A. Porretta, Entropy solutions for nonlinear elliptic equations in L1, Nonlinear Analysis, Theory, Methods & Applications, 32 (1998), 325-334. doi: 10.1016/S0362-546X(96)00323-9. [12] M. Liero, T. Koprucki, A. Fischer, R. Scholz and A. Glitzky, p-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices, Zeitschrift für Angewandte Mathematik und Physik, 66 (2015), 2957-2977. doi: 10.1007/s00033-015-0560-8. [13] P. Lindqvist, Notes on the p-Laplace Equation, Report 102, University of Jyväskyla, Department of Mathematics and Statistics, ISBN: 951-39-2586-2,2006. [14] L. Orsina, Elliptic equations with measure data, 2013, URL http://www1.mat.uniroma1.it/people/orsina/AS1213/AS1213.pdf. [15] T. Roubíček, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics, 153, Springer, Basel, 2013.
 [1] Joachim Naumann. On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 837-852. doi: 10.3934/dcdss.2017042 [2] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171 [3] Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587 [4] Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 [5] Yasir Ali, Arshad Alam Khan. Exact solution of magnetohydrodynamic slip flow and heat transfer over an oscillating and translating porous plate. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 595-606. doi: 10.3934/dcdss.2018034 [6] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [7] Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861 [8] Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 [9] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [10] Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919 [11] SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026 [12] Rinaldo M. Colombo, Mauro Garavello. Comparison among different notions of solution for the $p$-system at a junction. Conference Publications, 2009, 2009 (Special) : 181-190. doi: 10.3934/proc.2009.2009.181 [13] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715 [14] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [15] Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809 [16] Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021 [17] Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure & Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23 [18] Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013 [19] Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044 [20] Youcef Amirat, Kamel Hamdache. Weak solutions to stationary equations of heat transfer in a magnetic fluid. Communications on Pure & Applied Analysis, 2019, 18 (2) : 709-734. doi: 10.3934/cpaa.2019035

2017 Impact Factor: 0.561