Existence and global stability of traveling curved fronts in the Allen-Cahn equations

By H. Ninomiya, Masaharu Taniguchi. 2005

164 citations recorded by the source corpus.

Abstract This paper is concerned with existence and stability of traveling curved fronts for the Allen–Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/70023688e65c0892f3f017816f1862c6b191b449

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Existence and qualitative properties of multidimensional conical bistable fronts (2005, background): “…in dimensions N ≥ 3, the levels sets of any cylindrical solution u in Theorem 1.2 move away logarithmically from their asymptotic directions in dimensions N ≥ 3, in the sense that |φλ(x) + |x| cotα − k log |x| + tλ| = o(1) as |x| = |(x1, . . . , xN−1)| → +∞ for all λ ∈ (0, 1) and for some k > 0.”
Generalized Transition Waves and Their Properties (2010, background): “1) in R for usual types of nonlinearities f (combustion, bistable, Kolmogorov-Petrovsky-Piskunov type), see [3, 10, 17, 18, 21, 22, 31, 40, 41] for details.”
Periodic traveling curved fronts in reaction–diffusion equation with bistable time-periodic nonlinearity (2011, background): “Then a direct calculation yields (see also [17]) ζx =− αφ ′φ′′ 1+ φ′2 ζ + φ′ √ 1+ φ′2 , ζxx =− ′′2 + α2φ′φ′′′ 1+ φ′2 ζ + 3α2φ′2φ′′2 (1+ φ′2)2 ζ − α(φ′2 − 1)φ′′ (1+ φ′2)3/2 .”
Spreading and Vanishing for a Monostable Reaction–Diffusion Equation with Forced Speed (2018, background): “However, when c > c∗, we expect that there exist conical wave solutions, which converge locally uniformly to 0 as x → +∞ but not uniformly; see for instance [14,22] for details on conical waves.”
Asymptotic properties and classification of bistable fronts with Lipschitz level sets (2005, unknown): “{(r, ψ(r)), r ≥ 0}) the level sets {v = θ1} (resp.”
Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity (2012, background): “[37] H. Ninomiya, M. Taniguchi, Global stability of traveling curved fronts in the Allen–Cahn equations, Discrete Contin.”