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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

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
$u$
.
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. BoccardoA. 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. FischerP. PahnerB. LüssemK. LeoR. ScholzT. KopruckiJ. FuhrmannK. 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. HowisonJ. 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. LieroT. KopruckiA. FischerR. 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. ShawV. V. MitinE. 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. BoccardoA. 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. FischerP. PahnerB. LüssemK. LeoR. ScholzT. KopruckiJ. FuhrmannK. 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. HowisonJ. 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. LieroT. KopruckiA. FischerR. 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. ShawV. V. MitinE. 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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