A note on the thermistor problem in two space dimensions

Howison, Sam

doi:10.1090/qam/1012273

Cited by 27 publications

(13 citation statements)

References 3 publications

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“…We refer for more details to [6]. In the present situation there is always one and only one solution.…”

Section: Introductionmentioning

confidence: 97%

Asymptotics for the time-dependent thermistor problem

Cimatti¹

2004

Quart. Appl. Math.

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Abstract.Using the Liapunoff method, the global asymptotic stability of the stationary solution of the time-dependent thermistor problem is proved.In this paper we study a special case of the time-dependent thermistor problem, for which it is possible to give a fairly complete description of the asymptotic behaviour of the solution. A difference of potential V, in series with an ordinary resistor R, is applied to an indefinite slab of width d and electric conductivity a(u). We model this situation with the following one-dimensional initial-boundary value problem (D):where ip(x, t) is the electric potential and u{x, t) the temperature. In the usual thermistor problem, R is taken equal to 0, but this ignores e.g. the internal resistance of the generator.Thus, the present model is physically more realistic. The method is based on the use of a Liapunoff function. We treat in details the case a(u) = a0eu,which is relevant in the theory of PTC thermistor [5].In recent years, the thermistor problem has been the subject of many mathematical investigations. We quote, among others, the work of W. Allegretto, W. Lin ([1], [2]), and A. Lacey ([8], [9]) for the time-dependent case and, for the steady case, the papers of S.Howison, F. Rodrigues, and M. Shilor ([6], [7]).

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“…We refer for more details to [6]. In the present situation there is always one and only one solution.…”

Section: Introductionmentioning

confidence: 97%

Asymptotics for the time-dependent thermistor problem

Cimatti¹

2004

Quart. Appl. Math.

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“…In particular in [9] the invariance by conformal mappings of the problem is noted. The thermistor problem reads…”

Section: Introductionmentioning

confidence: 99%

Invariance of flows in doubly-connected domains with the same modulus

Cimatti

2014

Boll Unione Mat Ital

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We consider systems of elliptic partial differential equations in divergence form with Dirichlet's boundary conditions in doubly-connected domain of the plane with modulus μ. We prove an invariance property of the corresponding global flows in the class of domains with the same modulus. Applications are given to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour's effects.Keywords Systems of PDE in divergence form · Doubly-connected plane domain · Thermistor problem · Porous media · Darcy law · Soret effect · Dufour effect Mathematics Subject Classification 30E30 · 35J66 · 35J57

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“…Such materials are being used in switches and electric surge protecting devices, among other applications. It was brought to the Oxford Study Groups with Industry in the mid 80's and has received, as a result, considerable attention in the mathematical literature, see, e.g., [1,2,3,5,7,8,9,12,15,16,19,24,25,26,28] and references therein. Related problems can be found in [21] and [27,14].…”

Section: Introductionmentioning

confidence: 99%

Existence for the thermoviscoelastic thermistor problem

Kuttler

Shillor

Fernández

2008

Differ Equ Dyn Syst

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The existence of a weak solution to a dynamic model for a thermistor, which takes into account the thermoelastic properties of the device, is established. The model consists of a coupled system of the equations of dynamic thermoviscoelasticity, the heat equation with the Joule heating term, and the quasistatic charge conservation equation. The system is strongly nonlinear since the electrical conductivity is assumed to be temperature dependent, and the Joule heating term is quadratic in the gradient of the electric potential. The existence of a solution is obtained by considering a sequence of approximate time-retarded problems. After obtaining the necessary a priori estimates, a solution of the problem is found by passing to the approximation limit. The uniqueness of the solution remains an open problem. c 2008 Foundation for Scientific Research and Technological Innovation(FSRTI). All rights reserved.MSC: 74D05, 74F05, 74H20, 74H30

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A note on the thermistor problem in two space dimensions

Cited by 27 publications

References 3 publications

Asymptotics for the time-dependent thermistor problem

Asymptotics for the time-dependent thermistor problem

Invariance of flows in doubly-connected domains with the same modulus

Existence for the thermoviscoelastic thermistor problem

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