Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions

Cimatti, Giovanni

doi:10.1090/qam/987900

Cited by 98 publications

(60 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For another configuration of the conductor-thermistor, the so-called narrowing process, which considers that only a small part, and not the whole of the thermistor, is cylindrical, see [5]. Also, steady states of the full problem (1.1), (1.2) were investigated in [1,10,11]. Equation (1.3) can also be used to describe thermoviscous flow of linear materials.…”

Section: Introductionmentioning

confidence: 99%

On the Blow-Up of the Non-Local Thermistor Problem

Kavallaris

Nadzieja²

2007

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

The conditions under which the solution of the non-local thermistor problemblows up are investigated. We assume that f (s) is a decreasing function and that it is integrable in (0, ∞). Considering a suitable functional we prove that for all λ > 0 the solution of the Neumann problem blows up in finite time. The same result is obtained for the Robin problem under the assumption that λ is sufficiently large (λ 1). In the proof of existence of blow-up for the Dirichlet problem we use the subsolution technique. We are able to construct a blowing-up lower solution under the assumption that either λ > λ * or 0 < λ < λ * , for some critical value λ * , and that the initial condition is sufficiently large provided also that f (s) satisfies the decay condition

show abstract

Section: Introductionmentioning

confidence: 99%

On the Blow-Up of the Non-Local Thermistor Problem

Kavallaris

Nadzieja²

2007

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

show abstract

“…To our knowledge, there is no general result on the numerical analysis of problem (1.1). For mathematical and numerical analyses of simpler problems consisting of nonlinear coupled systems of two scalar elliptic equations, we refer to [6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Finite element analysis of a coupled thermally dependent viscosity flow problem

Zhu¹,

Loula²,

F.³

et al. 2007

Mat. apl. comput.

View full text Add to dashboard Cite

Abstract. In this work, a stationary Stokes flow with thermal effects is studied both mathematically and numerically. First, existence, uniqueness and regularity of the weak solution of the problem are established. Next, fi nite element approximation to the problem, based on a fi xed point algorithm, is proposed. Then, an error estimate between continuous solution and discrete one is obtained. Finally, some numerical tests are presented to confi rm the theoretical results.Mathematical subject classification: 35J, 35Q, 65N30, 76.

show abstract

“…For certain boundary conditions these equations may be reduced to one nonlinear o.d.e. and Laplace's equation, as noted in [1]. We give a geometrical interpretation of this reduction in two space dimensions in terms of a conformal map from the thermistor onto a rectangle.…”

mentioning

confidence: 96%

“…In a recent paper [1] Cimatti has considered the following boundary value problem for the static temperature u and electric potential 0 This problem has also been studied in [2,3] and references therein. It is shown in [1] under very general assumptions on the functions k(u) and a(u) that, provided above. This is found from (1.1) and (1.2) to be with u = uq at 0 = 4>i, 4>2; that is, u{cfr) is given implicitly by the formula…”

mentioning

confidence: 99%