Stefan problem with a kinetic condition at the free boundary

1986 ◽  
Vol 146 (1) ◽  
pp. 97-122 ◽  
Author(s):  
A. Visintin
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Kinetic Condition

Critical exponent in a Stefan problem with kinetic condition

1997 ◽  
Vol 8 (5) ◽  
pp. 525-532 ◽  
Author(s):  
ZHICHENG GUAN ◽  
XU-JIA WANG
Keyword(s):  
Free Boundary ◽  
Critical Exponent ◽  
Global Solution ◽  
Stefan Problem ◽  
Finite Time ◽  
Blow Up ◽  
One Dimensional ◽  
Kinetic Condition ◽  
Dirac Function ◽  
The One

In this paper we deal with the one-dimensional Stefan problemut−uxx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ+, u(x, 0) =u0(x)with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u)[ges ] Ceγ1[mid ]u[mid ] for some γ1 large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u)[ges ] Ce√2πe[mid ]u[mid ].

scholarly journals Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary

1989 ◽  
Vol 31 (1) ◽  
pp. 81-96 ◽  
Author(s):  
J. N. Dewynne ◽  
S. D. Howison ◽  
J. R. Ockendon ◽  
Weiqing Xie
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Large Time ◽  
Initial Temperature ◽  
Asymptotic Behavior Of Solutions ◽  
Infinite Strip ◽  
Time Behaviour ◽  
Large Time Behaviour ◽  
Kinetic Undercooling ◽  
Kinetic Condition

AbstractWe study the large time behaviour of the free boundary for a one-phase Stefan problem with supercooling and a kinetic condition u = −ε|⋅ṡ| at the free boundary x = s(t). The problem is posed on the semi-infinite strip [0,∞) with unit Stefan number and bounded initial temperature ϕ(x) ≤ 0, such that ϕ → −1 − δ as x → ∞, where δ is constant. Special solutions and the asymptotic behaviour of the free boundary are considered for the cases ε ≥ 0 with δ negative, positive and zero, respectively. We show that, for ε > 0, the free boundary is asymptotic to , δt/ε if < δ > 0 respectively, and that when δ = 0 the large time behaviour of the free boundary depends more sensitively on the initial temperature. We also give a brief summary of the corresponding results for a radially symmetric spherical crystal with kinetic undercooling and Gibbs-Thomson conditions at the free boundary.

Large time behavior of the solutions to a one-dimensional Stefan problem with a kinetic condition at the free boundary

2004 ◽  
Vol 15 (3) ◽  
pp. 297-313 ◽  
Author(s):  
D. HILHORST ◽  
F. ISSARD-ROCH ◽  
J. M. ROQUEJOFFRE
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Large Time ◽  
Travelling Wave ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Nonlocal Parabolic Equation ◽  
Self Similar Solution ◽  
Kinetic Condition ◽  
Self Similar

We consider a Stefan problem with a kinetic condition at the free boundary and prove the convergence of the solution as $t$ tends to infinity either to a travelling wave solution or to a self-similar solution. The key idea is to transform this problem into a problem for a single nonlocal parabolic equation which admits a comparison principle.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

scholarly journals On the pointwise jump condition at the free boundary in the 1-phase Stefan problem

2005 ◽  
Vol 4 (2) ◽  
pp. 357-366
Author(s):  
Donatella Danielli ◽  
◽  
Marianne Korten ◽  
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Jump Condition

Control of the free boundary in a two-phase Stefan problem with flow driven by convection

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 4140027-4140028
Author(s):  
Michael Hinze ◽  
Stefan Ziegenbalg
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Two Phase
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