A Bound for the Temperature in the Thermistor Problem

Temperature Surges in Current-Limiting Circuit Devices

“…The quadratic nonlinearity in the field equations has recently stimulated mathematical interest in this version of the problem, and results are available for various boundary conditions for u and cp. Cimatti and Prodi [5] and Cimatti [2], [4] obtain existence and regularity results when u and cp have Dirichlet boundary data; i.e., both are specified on the thermistor boundary. Howison, Rodrigues, and Shillor [9] also obtain regularity results with the radiation condition ( 2 .…”

The thermistor problem for conductivity which vanishes at large temperature

“…(1.3) For the physical background of the thermistor problem and some explicit solutions we refer to [1], [9], [10], [11], and the references therein. There has been recent mathematical interest in the problem in case o(u) is uniformly positive; see [2], [3], [4], [7], [8]. Cimatti and Prodi in [2] and Cimatti in [3] considered the Dirichlet boundary conditions for both (p and u and proved existence of a solution.…”

“…There has been recent mathematical interest in the problem in case o(u) is uniformly positive; see [2], [3], [4], [7], [8]. Cimatti and Prodi in [2] and Cimatti in [3] considered the Dirichlet boundary conditions for both (p and u and proved existence of a solution. In [4] Cimatti extended the existence result to the case where cp = <p°, u -u° onrD, r^cdQ, d1> n du n r -in\r -= 0, ^-=o onryv = dQ\ro.…”

“…An important observation by Diesselhorst [5] that the function 1 2 fu ds 1"2*+LW) ° '5) satisfies the equation V(it(«)V^) = 0 inQ, (1.6) plays a crucial role in the papers [3], [4], In the special case td = r, u r2, <p = <p,, u = on r(, u = u on rD, -+ yu = g0 on dQ\ro .…”

“…In this paper we are interested in the case where o(u) vanishes for large u, i.e., a{u) > 0 if u < u*, ct(m) = 0 if u > u* (1)(2)(3)(4)(5)(6)(7)(8)(9) for some constant u*. This provides a good approximation to the actual engineering model of thermistors, whereby the conductivity o(u) drops to nearly 0 beyond some critical temperature u*.…”