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June  2016, 36(6): 3011-3034. doi: 10.3934/dcds.2016.36.3011

Existence of solutions for a model of microwave heating

Pierluigi Colli 1, and  Luca Scarpa 2,

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

Received  June 2015 Revised  October 2015 Published  December 2015

This paper is concerned with a system of differential equations related to a circuit model for microwave heating, complemented by suitable initial and boundary conditions. A RLC circuit with a thermistor is representing the microwave heating process with temperature-induced modulations on the electric field. The unknowns of the PDE system are the absolute temperature in the body, the voltage across the capacitor and the electrostatic potential. Using techniques based on monotonicity arguments and sharp estimates, we can prove the existence of a weak solution to the initial-boundary value problem.
Keywords: global existence., Microwave heating, evolutionary system of partial differential equations, circuit model, weak solution.
Mathematics Subject Classification: Primary: 35G61, 34A10, 35D30, 35Q7.
Citation: Pierluigi Colli, Luca Scarpa. Existence of solutions for a model of microwave heating. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3011-3034. doi: 10.3934/dcds.2016.36.3011
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

G. Cimatti, Remark on the number of solutions in the thermistor problem, Matematiche (Catania), 66 (2011), 49-60.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control for the thermistor problem,, SIAM J. Control Optim., 48 (): 3449.  doi: 10.1137/080736259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. Scarpa, A doubly nonlinear evolution problem related to a model for microwave heating, Adv. Math. Sci. Appl., 24 (2014), 251-275.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

G. Cimatti, Remark on the number of solutions in the thermistor problem, Matematiche (Catania), 66 (2011), 49-60.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control for the thermistor problem,, SIAM J. Control Optim., 48 (): 3449.  doi: 10.1137/080736259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. Scarpa, A doubly nonlinear evolution problem related to a model for microwave heating, Adv. Math. Sci. Appl., 24 (2014), 251-275.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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