# American Institute of Mathematical Sciences

June  2013, 33(6): 2349-2368. doi: 10.3934/dcds.2013.33.2349

## Thermal runaway for a nonlinear diffusion model in thermal electricity

 1 Department of Mathematics, Sichuan University, Chengdu 610064, China 2 Department of Mathematics, Jingcheng College of Sichuan University, Chengdu 611731, China

Received  January 2012 Revised  May 2012 Published  December 2012

In this paper, we consider the phenomena of the thermal runaway and the asymptotic runaway in a nonlocal nonlinear model, which is raised from the thermal-electricity and it is so-called an Ohmic heating model. The model prescribes the dimensionless temperature when the electric current flows through two conductors, subject to a fixed potential difference. The electrical resistivity of the one of the conductors depends on the temperature and the other one remains constant. The problem will be mathematically formulated to a quasilinear nonlocal parabolic equation with Dirichlet boundary condition. An analysis of the problem shows that the solution of the problem exists globally, provided that the conductor with constant resistivity is connected. Furthermore, for some special temperature-resistivity relations, the unique stationary solution is shown to be global asymptotically stable. The results assert a physical fact that the thermal produced by the Ohmic heating process will runaway from the surfaces of the conductor, the temperature of the conductor remains bounded and solution of the system converges asymptotically to the unique equilibrium.
Citation: Lili Du, Mingshu Fan. Thermal runaway for a nonlinear diffusion model in thermal electricity. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2349-2368. doi: 10.3934/dcds.2013.33.2349
##### References:
 [1] D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients, Bull. Amer. Math. Soc., 69 (1963), 841-847. [2] J. W. Bebernes and R. Ely, Existence and invariance for parabolic functional equations, Nonlinear Anal. TMA., 7 (1983), 1225-1235. doi: 10.1016/0362-546X(83)90054-8. [3] J. W. Bebernes and D. Eberly, "Mathematical Problems From Combustion Theory," Applied Mathematical Sciences, 83, Springer-Verlag, New York, 1989. [4] J. W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal., 3 (1996), 79-103. [5] E. Dibenedetto, "Degenerate Parabolic Equations," Springer, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [6] L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247. doi: 10.1016/j.cam.2006.02.028. [7] L. Du, C. Mu and M. Fan, Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source, Dyn. Syst., 20 (2005), 401-412. doi: 10.1080/14689360500238818. [8] L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation, Commun. Pure Appl. Anal., 6 (2007), 183-190. doi: 10.3934/cpaa.2007.6.183. [9] M. Fan and L. Du, Asymptotic behavior for an Ohmic heating model in thermal electricity, Appl. Math. Comput., 218 (2012), 10906-10913. doi: 10.1016/j.amc.2012.04.053. [10] M. Fan, C. Mu, L. Du, Uniform blow-up profiles for a nonlocal degenerate parabolic system, Appl. Math. Sci., 1 (2007), 13-23. [11] N. I. Kavallaris, Asymptotic behaviour and blow-up for a nonlinear diffusion problem with a non-local source term, Proc. Edinb. Math. Soc., 47 (2004), 375-395. doi: 10.1017/S0013091503000658. [12] A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some spacial cases, Eur. J. Appl. Math., 6 (1995), 127-144. [13] A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part II: General proof of blow-up and asymptotics of runaway, Euro. J. Appl. Math., 6 (1995), 201-224. doi: 10.1017/S0956792500001807. [14] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Trans. of Math. Monographs, 23, 1968. [15] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States," Birkhäuser Verlag, Basel, 2007. [16] Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334. doi: 10.1137/S0036141097318900. [17] Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differ. Eqns, 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.

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##### References:
 [1] D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients, Bull. Amer. Math. Soc., 69 (1963), 841-847. [2] J. W. Bebernes and R. Ely, Existence and invariance for parabolic functional equations, Nonlinear Anal. TMA., 7 (1983), 1225-1235. doi: 10.1016/0362-546X(83)90054-8. [3] J. W. Bebernes and D. Eberly, "Mathematical Problems From Combustion Theory," Applied Mathematical Sciences, 83, Springer-Verlag, New York, 1989. [4] J. W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal., 3 (1996), 79-103. [5] E. Dibenedetto, "Degenerate Parabolic Equations," Springer, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [6] L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247. doi: 10.1016/j.cam.2006.02.028. [7] L. Du, C. Mu and M. Fan, Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source, Dyn. Syst., 20 (2005), 401-412. doi: 10.1080/14689360500238818. [8] L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation, Commun. Pure Appl. Anal., 6 (2007), 183-190. doi: 10.3934/cpaa.2007.6.183. [9] M. Fan and L. Du, Asymptotic behavior for an Ohmic heating model in thermal electricity, Appl. Math. Comput., 218 (2012), 10906-10913. doi: 10.1016/j.amc.2012.04.053. [10] M. Fan, C. Mu, L. Du, Uniform blow-up profiles for a nonlocal degenerate parabolic system, Appl. Math. Sci., 1 (2007), 13-23. [11] N. I. Kavallaris, Asymptotic behaviour and blow-up for a nonlinear diffusion problem with a non-local source term, Proc. Edinb. Math. Soc., 47 (2004), 375-395. doi: 10.1017/S0013091503000658. [12] A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some spacial cases, Eur. J. Appl. Math., 6 (1995), 127-144. [13] A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part II: General proof of blow-up and asymptotics of runaway, Euro. J. Appl. Math., 6 (1995), 201-224. doi: 10.1017/S0956792500001807. [14] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Trans. of Math. Monographs, 23, 1968. [15] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States," Birkhäuser Verlag, Basel, 2007. [16] Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334. doi: 10.1137/S0036141097318900. [17] Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differ. Eqns, 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.
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