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August  2017, 10(4): 837-852. doi: 10.3934/dcdss.2017042

On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities

Joachim Naumann

Department of Mathematics, Humboldt University Berlin, Unter den Linden 6,10099 Berlin, Germany

Received  April 2016 Revised  December 2016 Published  April 2017

Let
$\Omega\subset\mathbb{R}^n$
 (
$n=2$
 or
$n=3$
) be a bounded domain. We consider the thermistor system
$\text{(1)}\quad \nabla\cdot \boldsymbol{J}=0,\qquad \text{(2)}\quad \frac{\partial u}{\partial t}+\nabla\cdot\boldsymbol{q}=f(x,t,u,\nabla\varphi)\;\text{ in }\; \Omega\times\,]\,0,T\,[\,,$
where (1) is a
$p$
-Laplace type equation for
$\varphi$
 (
$u=$
 temperature,
$\varphi=$
 electrostatic potential). We prove the existence of a weak solution
$(\varphi,u)$
 of (1)–(2) under mixed boundary conditions for
$\varphi$
, and a Robin boundary condition and an initial condition for
$u$
.
Keywords: Thermistor system, Robin boundary condition, p-Laplacian.
Mathematics Subject Classification: 35K92, 35Q79, 80A20.
Citation: 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
References:
[1]

S. N. Antontsev and M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), 1128-1156.  doi: 10.1137/S0036141092233482.  Google Scholar

[2]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlin. Anal., T., M. & Appl., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.  Google Scholar

[3]

L. Boccardo, A. Dall'Aglio and T. Gallouët, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258.  doi: 10.1006/jfan.1996.3040.  Google Scholar

[4]

N. Bourbaki, Éléments de Mathématique, Livre VI, Intégration 1-4, Hermann, Paris, 1965. Google Scholar

[5]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publ. Comp. , Amsterdam, 1973. Google Scholar

[6]

F. E. Browder, Strongly nonlinear parabolic equation of higher order, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur., Rend. Lincei (9) Mat Appl., 77 (1986), 159-172.   Google Scholar

[7]

M. Bulíček, A. Glitzky and M. Liero, Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents, www.wias-berlin.de/publications/wias-publ./no.2206. Google Scholar

[8]

G. Cimatti, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quarterly Appl. Math., 47 (1989), 117-121.  doi: 10.1090/qam/987900.  Google Scholar

[9]

G. Cimatti, Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor, Ann. Mat. Pura Appl., 162 (1992), 33-42.  doi: 10.1007/BF01759998.  Google Scholar

[10]

G. Cimatti, The thermistor problem with Robin boundary condition, Rend. Semin. Mat. Univ. Padova, 135 (2016), 175-199.  doi: 10.4171/RSMUP/135-10.  Google Scholar

[11]

J. Droniou, Intégration et espaces de Sobolev à valeurs vectorielles, www-gm3.univ-mrs.fr/polys/. Google Scholar

[12]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc. , Providence, R. I. , 1998. doi: 10.1090/gsm/019.  Google Scholar

[13]

A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, T. Koprucki, J. Fuhrmann, 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.  Google Scholar

[14]

A. Glitzky and M. Liero, Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices. www.wias-berlin.de/publications/wias-publ./no.2143. Google Scholar

[15]

S. D. Howison, J. F. Rodrigues and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl., 174 (1993), 573-588.  doi: 10.1006/jmaa.1993.1142.  Google Scholar

[16]

K. A. Jenkins and K. Rim, Measurements of the effect of self-heating in strained-silicon MOSFETs, IEEE Electr. Device Lett., 23 (2002), 360-362.  doi: 10.1109/LED.2002.1004235.  Google Scholar

[17]

J Leray and J. -L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.   Google Scholar

[18]

M. Liero, T. Koprucki, A. Fischer, R. Scholz and A. Glitzky, p-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices, Z. Angew. Math. Phys., 66 (2015), 2957-2977.  doi: 10.1007/s00033-015-0560-8.  Google Scholar

[19]

P. Lindqvist, Notes on the p-Laplace Equation, Report. University of Jyväskylä, Department of Mathematics and Statistics, 102. Univ. Jyväskylä, Jyväskylä 2006, 80 pp. Google Scholar

[20]

J. -L. Lions, Quelques Méthodes de Résolution de Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar

[21]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Wien, New York, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[22]

J. Naumann, On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids, Math. Meth. Appl. Sci., 29 (2006), 1883-1906.  doi: 10.1002/mma.754.  Google Scholar

[23]

J. M. Rakotoson, Some quasilinear parabolic equations, Nonlin. Anal., T. M. & A., 17 (1991), 1163-1175.  doi: 10.1016/0362-546X(91)90235-S.  Google Scholar

[24]

J. M. Rakotoson, A compactness lemma for quasilinear problems: Application to parabolic equations, J. Funct. Anal., 106 (1992), 358-374.  doi: 10.1016/0022-1236(92)90053-L.  Google Scholar

[25] M. P. Shaw, V. V. Mitin, E. Schöll and H. L. Gubin, The Physics of Instabilities in Solid State Electron Devices, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4899-2344-8.  Google Scholar
[26]

J. Simon, Compact sets in the spaces Lp(0, T; B), Annali Mat. Pura Appl., 146 (1987), 65-96.   Google Scholar

[27]

S. M. Sze and K. Ng Kwok, Physics of Semiconductor Devices, 3rd ed. , J. Wiley, New Jersey, 2007. doi: 10.1063/1.3022205.  Google Scholar

[28]

X. Xu, A p-Laplacian problem in L1 with nonlinear boundary conditions, Comm. Partial Differential Equations, 19 (1994), 143-176.   Google Scholar

[29]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. Ⅱ/B: Nonlinear Monotone Operators, New York, Berlin, Springer-Verlag, 1990. Google Scholar

show all references

References:
[1]

S. N. Antontsev and M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), 1128-1156.  doi: 10.1137/S0036141092233482.  Google Scholar

[2]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlin. Anal., T., M. & Appl., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.  Google Scholar

[3]

L. Boccardo, A. Dall'Aglio and T. Gallouët, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258.  doi: 10.1006/jfan.1996.3040.  Google Scholar

[4]

N. Bourbaki, Éléments de Mathématique, Livre VI, Intégration 1-4, Hermann, Paris, 1965. Google Scholar

[5]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publ. Comp. , Amsterdam, 1973. Google Scholar

[6]

F. E. Browder, Strongly nonlinear parabolic equation of higher order, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur., Rend. Lincei (9) Mat Appl., 77 (1986), 159-172.   Google Scholar

[7]

M. Bulíček, A. Glitzky and M. Liero, Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents, www.wias-berlin.de/publications/wias-publ./no.2206. Google Scholar

[8]

G. Cimatti, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quarterly Appl. Math., 47 (1989), 117-121.  doi: 10.1090/qam/987900.  Google Scholar

[9]

G. Cimatti, Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor, Ann. Mat. Pura Appl., 162 (1992), 33-42.  doi: 10.1007/BF01759998.  Google Scholar

[10]

G. Cimatti, The thermistor problem with Robin boundary condition, Rend. Semin. Mat. Univ. Padova, 135 (2016), 175-199.  doi: 10.4171/RSMUP/135-10.  Google Scholar

[11]

J. Droniou, Intégration et espaces de Sobolev à valeurs vectorielles, www-gm3.univ-mrs.fr/polys/. Google Scholar

[12]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc. , Providence, R. I. , 1998. doi: 10.1090/gsm/019.  Google Scholar

[13]

A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, T. Koprucki, J. Fuhrmann, 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.  Google Scholar

[14]

A. Glitzky and M. Liero, Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices. www.wias-berlin.de/publications/wias-publ./no.2143. Google Scholar

[15]

S. D. Howison, J. F. Rodrigues and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl., 174 (1993), 573-588.  doi: 10.1006/jmaa.1993.1142.  Google Scholar

[16]

K. A. Jenkins and K. Rim, Measurements of the effect of self-heating in strained-silicon MOSFETs, IEEE Electr. Device Lett., 23 (2002), 360-362.  doi: 10.1109/LED.2002.1004235.  Google Scholar

[17]

J Leray and J. -L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.   Google Scholar

[18]

M. Liero, T. Koprucki, A. Fischer, R. Scholz and A. Glitzky, p-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices, Z. Angew. Math. Phys., 66 (2015), 2957-2977.  doi: 10.1007/s00033-015-0560-8.  Google Scholar

[19]

P. Lindqvist, Notes on the p-Laplace Equation, Report. University of Jyväskylä, Department of Mathematics and Statistics, 102. Univ. Jyväskylä, Jyväskylä 2006, 80 pp. Google Scholar

[20]

J. -L. Lions, Quelques Méthodes de Résolution de Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar

[21]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Wien, New York, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[22]

J. Naumann, On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids, Math. Meth. Appl. Sci., 29 (2006), 1883-1906.  doi: 10.1002/mma.754.  Google Scholar

[23]

J. M. Rakotoson, Some quasilinear parabolic equations, Nonlin. Anal., T. M. & A., 17 (1991), 1163-1175.  doi: 10.1016/0362-546X(91)90235-S.  Google Scholar

[24]

J. M. Rakotoson, A compactness lemma for quasilinear problems: Application to parabolic equations, J. Funct. Anal., 106 (1992), 358-374.  doi: 10.1016/0022-1236(92)90053-L.  Google Scholar

[25] M. P. Shaw, V. V. Mitin, E. Schöll and H. L. Gubin, The Physics of Instabilities in Solid State Electron Devices, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4899-2344-8.  Google Scholar
[26]

J. Simon, Compact sets in the spaces Lp(0, T; B), Annali Mat. Pura Appl., 146 (1987), 65-96.   Google Scholar

[27]

S. M. Sze and K. Ng Kwok, Physics of Semiconductor Devices, 3rd ed. , J. Wiley, New Jersey, 2007. doi: 10.1063/1.3022205.  Google Scholar

[28]

X. Xu, A p-Laplacian problem in L1 with nonlinear boundary conditions, Comm. Partial Differential Equations, 19 (1994), 143-176.   Google Scholar

[29]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. Ⅱ/B: Nonlinear Monotone Operators, New York, Berlin, Springer-Verlag, 1990. Google Scholar

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