The thermistor problem with conductivity vanishing for large temperature

Abstract

We consider the system (d/dt)u = Aw + cr(u)\V<p\ 2 , div (<r(u)Vp) = 0 in a bounded region of WL" coupled with initial and boundary conditions, where <j(s) e C(R) is nonnegative and <J(U) = 0 if and only if u S: a for some a > 0. Owing to the degeneracy involved, solutions of the problem display new phenomena that cannot be incorporated into the classical weak formulation. The notion of a capacity solution introduced in [14,15] is employed to study the problem. It turns out that this notion of a solution is just general enough to encompass the new phenomena involved.

Existence for the thermoviscoelastic thermistor problem

“…In some publications (see, e.g., [25,26] and references therein) the degenerate case was investigated, where σ el was assumed to vanish above a critical temperature. This is an idealization of the material behavior which introduces considerable mathematical difficulties, and will not be pursued in this work.…”

“…Indeed, when σ = 0 the heating term in (2.1) vanishes and the equation for φ, (2.2), degenerates. This makes it necessary to consider the so-called capacity solutions of the problem, and we refer the reader to [25,26] and references therein for further details. …”

“…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].…”

“…Our interest lies in the general problem, and we assume that the electrical conductivity of the material does not vanish, and thus, the mathematical problem is nondegenerate. In the publications ( [14,24,25,26,27]) the degenerate problem has been investigated, which makes it necessary to use the so-called 'capacity solutions' that take into account the vanishing of the electrical conductivity. This, in turn, causes the degeneration of the elliptic equation.…”

Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

“…, f d ) represents the density of body forces. Moreover, if we denote by ε the linearized strain tensor (ε = ε(u) = Although different possibilities for σ el (θ) have been considered in many publications (including the so-called "capacity solutions" in [8,[22][23][24]), here it is assumed that the electrical conductivity is nondegenerate and bounded. Thus,…”