Asymptotics for the time-dependent thermistor problem

Abstract

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]).

A Projection-Based Error Analysis of HDG Methods for a Partially Coupled Elliptic Problem

“…the error convergence in 𝐻(div; 𝛺) and 𝐿 2 (𝛺) with order 𝑘 e 𝑘 + 1 respectively. This work makes an innovative numerical analysis of the thermistor model, which has been extensively studied, with a considerable number of works in the literature (Liero, 2015), (Loula, 2021), (Zhu, 2006), (Keskin, 2004), (Cimatti, 1995(Cimatti, , 2002(Cimatti, , 2004(Cimatti, , 2011, (Howison, 1993) and others. An innovative numerical analysis seen for the first time in the article by (Cockburn, 2010), was applied in a model of two coupled equations, and therefore some additional difficulties arise in relation to the application of the analysis of the model that was proposed in Cockbunr's work mentioned above.…”