# American Institute of Mathematical Sciences

November  2001, 1(4): 485-494. doi: 10.3934/dcdsb.2001.1.485

## On Maxwell's system with a thermal effect

 1 Department of Mathematics, Texas A & M University, College Station, Texas, 77843-3368, United States 2 Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 9916, United States

Received  February 2001 Revised  July 2001 Published  September 2001

In this paper we study Maxwell’s system coupled with a heat equation in one space dimension. The system models a microwave heating process. The feature of the model is that the electric conductivity $\sigma(u)$ strongly depends on the temperature. It is shown that the system has a global solution for $\sigma(u)=1+u^k$ with any $k\ge 1$. The long time behavior of the solution is also investigated.
Citation: J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485
 [1] Pierluigi Colli, Luca Scarpa. Existence of solutions for a model of microwave heating. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3011-3034. doi: 10.3934/dcds.2016.36.3011 [2] Asim Aziz, Wasim Jamshed, Yasir Ali, Moniba Shams. Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2667-2690. doi: 10.3934/dcdss.2020142 [3] Yumei Liao, Wei Wei, Xianbing Luo. Existence of solution of a microwave heating model and associated optimal frequency control problems. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2103-2116. doi: 10.3934/jimo.2019045 [4] Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic and Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481 [5] W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431 [6] Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407 [7] S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590 [8] Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083 [9] Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations and Control Theory, 2019, 8 (2) : 343-357. doi: 10.3934/eect.2019018 [10] Wei Lv, Ruirui Sui. Optimality of piecewise thermal conductivity in a snow-ice thermodynamic system. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 47-57. doi: 10.3934/naco.2015.5.47 [11] V. S. Manoranjan, Hong-Ming Yin, R. Showalter. On two-phase Stefan problem arising from a microwave heating process. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1155-1168. doi: 10.3934/dcds.2006.15.1155 [12] Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867 [13] Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227 [14] Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 [15] Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879 [16] Miroslav Bartušek. Existence of noncontinuable solutions of a system of differential equations. Conference Publications, 2009, 2009 (Special) : 54-59. doi: 10.3934/proc.2009.2009.54 [17] Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91 [18] Yuri Kalinin, Volker Reitmann, Nayil Yumaguzin. Asymptotic behavior of Maxwell's equation in one-space dimension with thermal effect. Conference Publications, 2011, 2011 (Special) : 754-762. doi: 10.3934/proc.2011.2011.754 [19] Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257 [20] María Teresa González Montesinos, Francisco Ortegón Gallego. The evolution thermistor problem with degenerate thermal conductivity. Communications on Pure and Applied Analysis, 2002, 1 (3) : 313-325. doi: 10.3934/cpaa.2002.1.313

2020 Impact Factor: 1.327