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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. BoccardoT. 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.  Google Scholar

[2]

M. BulíčekA. 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.  Google Scholar

[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.  Google Scholar

[4]

G. Cimatti, Remark on the thermistor problem with rapidly growing conductivity, Applicable Analysis, 80 (2007), 133-140.  doi: 10.1080/00036810108840985.  Google Scholar

[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.  Google Scholar

[6]

A. FischerT. KopruckiK. GärtnerJ. BrücknerB. LüssemK. LeoA. 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.  Google Scholar

[7]

A. FischerP. PahnerB. LüssemK. LeoR. ScholzT. KopruckiK. 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.  Google Scholar

[8]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1975. doi: 10.1002/mana.19750672207.  Google Scholar

[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.  Google Scholar

[10]

A. Kufner, O. John and S. Fučik, Function Spaces, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977. Google Scholar

[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.  Google Scholar

[12]

M. LieroT. KopruckiA. FischerR. 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.  Google Scholar

[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. Google Scholar

[14]

L. Orsina, Elliptic equations with measure data, 2013, URL http://www1.mat.uniroma1.it/people/orsina/AS1213/AS1213.pdf. Google Scholar

[15]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics, 153, Springer, Basel, 2013. Google Scholar

show all references

References:
[1]

L. BoccardoT. 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.  Google Scholar

[2]

M. BulíčekA. 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.  Google Scholar

[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.  Google Scholar

[4]

G. Cimatti, Remark on the thermistor problem with rapidly growing conductivity, Applicable Analysis, 80 (2007), 133-140.  doi: 10.1080/00036810108840985.  Google Scholar

[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.  Google Scholar

[6]

A. FischerT. KopruckiK. GärtnerJ. BrücknerB. LüssemK. LeoA. 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.  Google Scholar

[7]

A. FischerP. PahnerB. LüssemK. LeoR. ScholzT. KopruckiK. 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.  Google Scholar

[8]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1975. doi: 10.1002/mana.19750672207.  Google Scholar

[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.  Google Scholar

[10]

A. Kufner, O. John and S. Fučik, Function Spaces, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977. Google Scholar

[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.  Google Scholar

[12]

M. LieroT. KopruckiA. FischerR. 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.  Google Scholar

[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. Google Scholar

[14]

L. Orsina, Elliptic equations with measure data, 2013, URL http://www1.mat.uniroma1.it/people/orsina/AS1213/AS1213.pdf. Google Scholar

[15]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics, 153, Springer, Basel, 2013. Google Scholar

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