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Existence of solutions for a model of microwave heating
1. | Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia |
2. | Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom |
References:
[1] |
S. Agrawal and G. A. Kriegsmann, A model for the microwave heating of a thin ceramic slab in a multimode cavity, IMA J. Appl. Math., 78 (2013), 652-664.
doi: 10.1093/imamat/hxt013. |
[2] |
S. N. Antonsev and M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), 1128-1156.
doi: 10.1137/S0036141092233482. |
[3] |
M. Badii, Existence of periodic solutions for the quasi-static thermoelastic thermistor problem, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 1-15.
doi: 10.1007/s00030-008-7017-0. |
[4] |
F. Brezzi and G. Gilardi, Chapters 1-3, in Finite Element Handbook (eds. H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987. |
[5] |
G. Cimatti, Remark on the number of solutions in the thermistor problem, Matematiche (Catania), 66 (2011), 49-60. |
[6] |
G. Cimatti, Remarks on the existence, uniqueness and semi-explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 481-495.
doi: 10.1017/S0308210509001826. |
[7] |
L. Consiglieri, A limit model for thermoelectric equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 57 (2011), 229-244.
doi: 10.1007/s11565-011-0129-1. |
[8] |
E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.
doi: 10.1002/mma.1089. |
[9] |
C. García Reimbert, M. C. Jorge, A. A. Minzoni and C. A. Vargas, Temperature modulations in a circuit model of microwave heating. Applied-mathematical perspectives on microwave processing, J. Engrg. Math., 44 (2002), 199-206.
doi: 10.1023/A:1020824314529. |
[10] |
D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control for the thermistor problem, SIAM J. Control Optim., 48 (2009/10), 3449-3481.
doi: 10.1137/080736259. |
[11] |
D. Hömberg and E. Rocca, A model for resistance welding including phase transitions and Joule heating, Math. Methods Appl. Sci., 34 (2011), 2077-2088.
doi: 10.1002/mma.1505. |
[12] |
K. L. Kuttler, M. Shillor and J. R. Fernández, Existence for the thermoviscoelastic thermistor problem, Differ. Equ. Dyn. Syst., 17 (2009), 217-233.
doi: 10.1007/s12591-009-0017-7. |
[13] |
V. S. Manoranjan, H.-M. Yin and R. Showalter, On two-phase Stefan problem arising from a microwave heating process, Discrete Contin. Dyn. Syst., 15 (2006), 1155-1168.
doi: 10.3934/dcds.2006.15.1155. |
[14] |
L. Scarpa, A doubly nonlinear evolution problem related to a model for microwave heating, Adv. Math. Sci. Appl., 24 (2014), 251-275. |
[15] |
P. Shi, M. Shillor and X. Xu, Existence of a solution to the Stefan problem with Joule's heating, J. Differential Equations, 105 (1993), 239-263.
doi: 10.1006/jdeq.1993.1089. |
[16] |
A. Sidi, R. Moulay and D. F. M. Torres, Optimal control of nonlocal thermistor equations, Internat. J. Control, 85 (2012), 1789-1801.
doi: 10.1080/00207179.2012.703789. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[18] |
W. Wei, H.-M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Anal., 75 (2012), 2024-2036.
doi: 10.1016/j.na.2011.10.003. |
[19] |
H.-M. Yin, Regularity of weak solution to Maxwell's equations and applications to microwave heating, J. Differential Equations, 200 (2004), 137-161.
doi: 10.1016/j.jde.2004.01.010. |
show all references
References:
[1] |
S. Agrawal and G. A. Kriegsmann, A model for the microwave heating of a thin ceramic slab in a multimode cavity, IMA J. Appl. Math., 78 (2013), 652-664.
doi: 10.1093/imamat/hxt013. |
[2] |
S. N. Antonsev and M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), 1128-1156.
doi: 10.1137/S0036141092233482. |
[3] |
M. Badii, Existence of periodic solutions for the quasi-static thermoelastic thermistor problem, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 1-15.
doi: 10.1007/s00030-008-7017-0. |
[4] |
F. Brezzi and G. Gilardi, Chapters 1-3, in Finite Element Handbook (eds. H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987. |
[5] |
G. Cimatti, Remark on the number of solutions in the thermistor problem, Matematiche (Catania), 66 (2011), 49-60. |
[6] |
G. Cimatti, Remarks on the existence, uniqueness and semi-explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 481-495.
doi: 10.1017/S0308210509001826. |
[7] |
L. Consiglieri, A limit model for thermoelectric equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 57 (2011), 229-244.
doi: 10.1007/s11565-011-0129-1. |
[8] |
E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.
doi: 10.1002/mma.1089. |
[9] |
C. García Reimbert, M. C. Jorge, A. A. Minzoni and C. A. Vargas, Temperature modulations in a circuit model of microwave heating. Applied-mathematical perspectives on microwave processing, J. Engrg. Math., 44 (2002), 199-206.
doi: 10.1023/A:1020824314529. |
[10] |
D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control for the thermistor problem, SIAM J. Control Optim., 48 (2009/10), 3449-3481.
doi: 10.1137/080736259. |
[11] |
D. Hömberg and E. Rocca, A model for resistance welding including phase transitions and Joule heating, Math. Methods Appl. Sci., 34 (2011), 2077-2088.
doi: 10.1002/mma.1505. |
[12] |
K. L. Kuttler, M. Shillor and J. R. Fernández, Existence for the thermoviscoelastic thermistor problem, Differ. Equ. Dyn. Syst., 17 (2009), 217-233.
doi: 10.1007/s12591-009-0017-7. |
[13] |
V. S. Manoranjan, H.-M. Yin and R. Showalter, On two-phase Stefan problem arising from a microwave heating process, Discrete Contin. Dyn. Syst., 15 (2006), 1155-1168.
doi: 10.3934/dcds.2006.15.1155. |
[14] |
L. Scarpa, A doubly nonlinear evolution problem related to a model for microwave heating, Adv. Math. Sci. Appl., 24 (2014), 251-275. |
[15] |
P. Shi, M. Shillor and X. Xu, Existence of a solution to the Stefan problem with Joule's heating, J. Differential Equations, 105 (1993), 239-263.
doi: 10.1006/jdeq.1993.1089. |
[16] |
A. Sidi, R. Moulay and D. F. M. Torres, Optimal control of nonlocal thermistor equations, Internat. J. Control, 85 (2012), 1789-1801.
doi: 10.1080/00207179.2012.703789. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[18] |
W. Wei, H.-M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Anal., 75 (2012), 2024-2036.
doi: 10.1016/j.na.2011.10.003. |
[19] |
H.-M. Yin, Regularity of weak solution to Maxwell's equations and applications to microwave heating, J. Differential Equations, 200 (2004), 137-161.
doi: 10.1016/j.jde.2004.01.010. |
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