An $(N-1)$-Dimensional Convex Compact Set Gives an $N$-Dimensional Traveling Front in the Allen--Cahn Equation | Zendy

Masaharu Taniguchi | Zendy

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An $(N-1)$-Dimensional Convex Compact Set Gives an $N$-Dimensional Traveling Front in the Allen--Cahn Equation

Author(s) -

Masaharu Taniguchi

Publication year - 2015

Publication title -

siam journal on mathematical analysis

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.882

H-Index - 92

eISSN - 1095-7154

pISSN - 0036-1410

DOI - 10.1137/130945041

Subject(s) - mathematics , allen–cahn equation , equivalence relation , regular polygon , mathematical analysis , perturbation (astronomy) , front (military) , equivalence (formal languages) , compact space , principal curvature , pure mathematics , mathematical physics , geometry , curvature , mean curvature , physics , quantum mechanics , meteorology

This paper studies traveling fronts to the Allen–Cahn equation in RN for N ≥ 3. Let(N −2)-dimensional smooth surfaces be the boundaries of compact sets in RN−1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation

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