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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.

[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.

[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.

[4]

N. Bourbaki, Éléments de Mathématique, Livre VI, Intégration 1-4, Hermann, Paris, 1965.
[5]

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

[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.
[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.

[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.

[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.

[11]

J. Droniou, Intégration et espaces de Sobolev à valeurs vectorielles, www-gm3.univ-mrs.fr/polys/.
[12]

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

[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.

[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.
[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.

[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.

[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.

[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.

[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.
[20]

J. -L. Lions, Quelques Méthodes de Résolution de Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
[21]

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

[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.

[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.

[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.

[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.
[26]

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

[27]

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

[28]

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

[29]

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

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.

[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.

[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.

[4]

N. Bourbaki, Éléments de Mathématique, Livre VI, Intégration 1-4, Hermann, Paris, 1965.
[5]

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

[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.
[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.

[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.

[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.

[11]

J. Droniou, Intégration et espaces de Sobolev à valeurs vectorielles, www-gm3.univ-mrs.fr/polys/.
[12]

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

[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.

[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.
[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.

[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.

[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.

[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.

[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.
[20]

J. -L. Lions, Quelques Méthodes de Résolution de Problèmes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
[21]

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

[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.

[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.

[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.

[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.
[26]

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

[27]

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

[28]

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

[29]

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