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Title: Spreading Profile and Nonlinear Stefan Problems
Contributor(s): Du, Yihong  (author)
Publication Date: 2013
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Abstract: We report some recent progress on the study of the following nonlinear Stefan problem ut − ∆u = f(u) for x ∈ Ω(t), t > 0, u = 0 and ut = µ|∇xu| 2 for x ∈ Γ(t), t > 0, u(0, x) = u0(x) for x ∈ Ω0, where Ω(t) ⊂ R N (N ≥ 1) is bounded by the free boundary Γ(t), with Ω(0) = Ω0, µ is a given positive constant. The initial function u0 is positive in Ω0 and vanishes on ∂Ω0. The class of nonlinear functions f(u) includes the standard monostable, bistable and combustion type nonlinearities. When µ → ∞, it can be shown that this free boundary problem converges to the corresponding Cauchy problem ut − ∆u = f(u) for x ∈ R N , t > 0, u(0, x) = u0(x) for x ∈ R N . We will discuss the similarity and differences of the dynamical behavior of these two problems by closely examining their spreading profiles, which suggest that the Stefan condition is a stabilizing factor in the spreading process.
Publication Type: Journal Article
Grant Details: ARC/DP120100727
Source of Publication: Bulletin of the Institute of Mathematics: Academia Sinica (New Series), 8(4), p. 413-430
Publisher: Institute of Mathematics, Academia Sinica
Place of Publication: Taiwan, Republic of China
ISSN: 2304-7895
Field of Research (FOR): 010110 Partial Differential Equations
Socio-Economic Outcome Codes: 970105 Expanding Knowledge in the Environmental Sciences
970101 Expanding Knowledge in the Mathematical Sciences
HERDC Category Description: C2 Non-Refereed Article in a Scholarly Journal
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