Stability of traveling curved fronts in a curvature flow with driving force

Abstract

Abstract. This paper is concerned with asymptotical stability of traveling curved fronts to a mean curvature flow with a constant driving force term in the two-dimensional Euclidean space. Our first result shows that, if a suitable bounded perturbation is added to a traveling curved front, it does not recover its shape at any positive time. This fact implies that boundedness of given perturbations is not enough for asymptotical stability. Then we prove that, if a given bounded perturbation decays at infinity, the perturbed traveling curved fronts always recover their shapes as time goes on. This fact holds true for any large perturbation if it decays at infinity. Introduction.Pattern formation is one of the most attractive fields in applied mathematics. Since traveling waves play important roles on the pattern formation, many researchers have studied them. Traveling waves in an one-dimensional media or planar traveling fronts in the plane seem to be mainly investigated so far. Fife [6] studied corner layers in the Allen-Cahn dynamics, which gives a first step to study traveling waves with more complicated shapes.This paper is concerned with the asymptotical stability of traveling curved fronts for a curvature flow with a constant driving force term in R 2 . Let Define r(t) = dD(i). Let 1/ be the normal vector on r(t) pointing from D(t) to D(t)c . The curvature H is given by H = -divi*. A pair of

Stability of travelling waves in a model for conical flames in two space dimensions

“…Let us mention here similar stability results were obtained by Ninomiya and Taniguchi [32] for curved fronts in singular limits for Allen-Cahn bistable equations. Existence of smooth solutions of problem (1.1-1.2) with bistable nonlinearity f was obtained by Fife [16] for angles α < π/2 close to π/2.…”

The self-similar expanding curve for the curvature flow equation

“…Then the behavior of the interface can be studied. In other methods, one seeks for a parametrization [19], graph representation [1,2,15,18,20,21], or auxiliary function [7] of the interface and then studies some alternative equations coming from (1.1).…”

“…In [18], Kohsaka considered one-dimensional graphs evolving by a general class of quasilinear equations on a domain with one free end. In [20,21], Ninomiya and Taniguchi considered one-dimensional graph representations of interfaces on the whole real line for F (κ, ν) = κ + c, where c is a nonzero constant. They focused on the so-called traveling curved fronts, such as the V-shaped waves which have been observed in many experiments and obtained various existence and asymptotic convergence results.…”

Traveling curved fronts of anisotropic curvature flows

“…q By the supersolutions and subsolutions in Lemma 4.3 we can show the local asymptotic stability of the traveling curved fronts by the similar argument to [13] which is based on [5,6]. THEOREM 4.4. Assume that u0( x) satisfies lim uo(x) -xtan(8+ -7r/2) = 0, (4.18)…”

“…Next we consider the asymptotic stability of the traveling curved front. As in [13], we search for a supersolution and a subsolution of the following type:…”