https://hdl.handle.net/1959.11/31913
Title: | Semi-waves with Lambda-shaped free boundary for nonlinear Stefan problems: Existence | Contributor(s): | Du, Yihong (author) ; Gui, Changfeng (author); Wang, Kelei (author); Zhou, Maolin (author) | Publication Date: | 2021 | Early Online Version: | 2021-03-02 | DOI: | 10.1090/proc/15346 | Handle Link: | https://hdl.handle.net/1959.11/31913 | Abstract: |
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{ | ut -∆ u = f(u), u > 0 | for x ∈ Ω(t) ⊂ ℝn+1, |
u = 0 and ut = μ | ∇xu|2 | for x ∈ Ωt, |
has a traveling wave solution whose free boundary is Λ-shaped, and whose speed is κ, where κ can be any given positive number satisfying κ > κ*, and κ* is the unique speed for which the above Stefan problem has a planar traveling wave solution. To distinguish it from the usual traveling wave solutions, we call it a semi-wave solution. In particular, if α ∈ (0, π/2) is determined by sin α = κ(*/)κ, then for any finite set of unit vectors (νi : 1 <= i <= m} ⊂ ℝn, there is a Λ-shaped semi-wave with traveling speed κ, with traveling direction -en+1n+1 , and with free boundary given by a hypersurface in ℝn+1 of the form
xn+1 = π(x1, ..., xn) = ɸ* (x1, ..., xn) + O(1) as |(x1, ..., xn)| -> ∞,
where
ɸ* (x1, ..., xn) \ colonequals - [max(1 <= i <= m) νi . (x1, ..., xn)] cot α
is a solution of the eikonal equation | ∇ ɸ | = cot α on ℝn.
0002-9939
School of Science and Technology
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