Solutions of Semilinear Elliptic Equations in with Conical&Shaped Level Sets

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

“…Furthermore, it follows from Theorem 1.7 in [19] that the solutions (c, φ) of (1.1-1.2) are unique, in the sense that c is unique, and φ up to a translation in (x, y). The speed c is necessarily equal to c = c 0 / sin α.…”

“…It was also proved in [19] that no solution of (1.1-1.2) exists whenever α ∈ (π/2, π), in dimensions 2 and higher.…”

“…Remark 1.2 1. The dimension 2 is quite different from other dimensions since, as soon as N ≥ 3, there is no solution of problem (1.1) in IR N , with α < π/2 and conical conditions of the type (1.2) (see [19]). But the possible existence of solutions of (1.1) in IR N under some weaker conical conditions is still open in dimensions N ≥ 3.…”

“…In the two-dimensional case with α = π/2, the existence of solutions φ of (1.1-1.2) was proved by Hamel and Monneau [19] for some angles α ∈ (0, π/2) and some functions f satisfying (1.4) under some additional assumptions (see Theorem 1.8 in [19]). Existence of solutions of (1.1) under some conical conditions weaker than (1.2) was also proved by Bonnet and Hamel (see Theorem 1.1 in [9]).…”

“…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.…”