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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 |
$\Omega\subset\mathbb{R}^n$ |
$n=2$ |
$n=3$ |
$\text{(1)}\quad \nabla\cdot \boldsymbol{J}=0,\qquad \text{(2)}\quad \frac{\partial u}{\partial t}+\nabla\cdot\boldsymbol{q}=f(x,t,u,\nabla\varphi)\;\text{ in }\; \Omega\times\,]\,0,T\,[\,,$ |
$p$ |
$\varphi$ |
$u=$ |
$\varphi=$ |
$(\varphi,u)$ |
$\varphi$ |
$u$ |
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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