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

Abstract

Abstract. This paper deals with the question of the stability of conical-shaped solutions of a class of reaction-diffusion equations in IR 2 . One first proves the existence of travelling waves solutions with conical-shaped level sets, generalizing earlier results by Bonnet, Hamel and Monneau [9], [19]. One then gives a characterization of the global attractor of these semilinear parabolic equations under some conical asymptotic conditions. Lastly, the global stability of the travelling waves solutions is proved.

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

“…Because Lip(φ n ; B(0, n)) ≤C, we see that up to a subsequence, (φ n ) n∈N * converges locally uniformly on R N −1 to φ 0 a viscosity solution to (10). Moreover, (ν n ) n∈N * converges to ν 0 ∈ S N −2 with…”

“…We first show the direct implication. Let φ ∞ ∈ C(R N −1 ) be a viscosity solution to (10). We shall prove that φ ∞ is (cot α)-Lipschitz and concave before giving its characterisation as an infimum of affine maps.…”

“…Step 5: The 1-homogeneous case We assume that φ ∞ is a 1-homogeneous continuous viscosity solution to (10). Then for any x 0 ∈ R N −1 , there exists p ∈ R N −1 with |p| = cot α such that by (11)…”

“…Paper [10] proves the existence of conical travelling waves solutions to (4), and paper [11] classifies all possible bounded non constant travelling waves solutions under rather weak conditions at infinity. In particular, it is proved in [11] that c ≥ c 0 and, up to a shift in x ∈ R, either u is a planar front φ 0 (±x cos α + z sin α) with α = arcsin(c 0 /c) ∈ (0,…”

“…In a second one, we reduce our study to 1-homogeneous functions and give a better description of those solutions in order to use them in both sections 4 and 5. (10) in any dimension N For any unit vector ν ∈ S N −2 and γ ∈ (−∞, +∞], let us define the affine map …”