The thermistor problem for conductivity which vanishes at large temperature

Abstract

Abstract. The thermistor problem is modeled as a coupled system of nonlinear elliptic equations. When the conductivity coefficient a(u) vanishes (u = temperature) one of the equations becomes degenerate; this situation is considered in the present paper. We establish the existence of a weak solution and, under some special Dirichlet and Neumann boundary conditions, analyze the structure of the set {a(u) = 0} and also prove uniqueness.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

“…Work related to this model can be found elsewhere [2,5,6,7,8,11]. Estimates of this type are very important since they answer to the question 'when' the blow up takes place [3,13].…”

The thermistor problem with conductivity vanishing for large temperature

“…The approach adopted here is different from that in [2,3]. Our assumptions on the data are much weaker.…”

“…In this paper we consider the case where a(s) is continuous and satisfies o(s) = 0 for s 2: a, a > 0 and 0 < a(s) <M for -oo < s < a, M > 0. (1.5) This situation arises in connection with the study of the combined processes of heat conduction and electrical conduction in a thermistor [2,3]. Then a is the critical temperature value.…”

Application of the Front-Fixing Method to Numerical Modeling of Field Diffusion in Nonlinear Systems

“…The transition region for a given point between temperatures and , which corresponds to coordinates and , is typically small and a coupled treatment of electric and thermal fields is complex. It is quite common to ignore the transition and consider a sharp interface with a step behavior in [3], [4]. But it was noted [2], [5] that the step function is not the most realistic model for and that a more complex relationship should be used for accurate predictions.…”