Periodic travelling wave solutions of a parabolic equation: a monotonicity result

Traveling waves of a curvature flow in almost periodic media

“…This also implies that there is a unique solution to (9). We denote the solution by (c n , p n , ϕ n ).…”

“…Next we apply some of the ideas in [8,9] showing that the average of tan ϕ (or any f (ϕ) where f is increasing) on the interval [−n, n] is a monotonic function of p. Thus, for each fixed α, there is a unique p, such that this average is exactly tan α. We denote this solution (with α fixed) by (c n , ϕ n ).…”

“…Let α ∈ (−π /2, π/2) be given. For each n > 0, problem (9), for unknown (c, p, ϕ) ∈ R × (−π /2, π/2) × C 1 ([−n, n]), admits a unique solution. Denote the solution by (c n , p n , ϕ n ).…”

“…Let α ∈ (−π /2, π/2) be fixed. For each n ∈ N let (c n , ϕ n (x)) be the unique solution of(9). Then there exist a subsequence {n i } of {n}, c > 0 and ϕ(x) ∈ C 1 (R) such that, as i → ∞, c n i → c, and…”

Homogenization of a semilinear heat equation

“…Notice that, since g depends only on u, these pulsating waves are in fact traveling waves which moves horizontally in the p-direction. Such solutions are related to the correctors used in homogenization problems and are very important in the analysis of long time behavior of the solutions to (1), with ε = 1, since typically they are the long time attractors of such solutions, see for instance [6,7,10,11,9,19]. In particular in [19], it is proved the existence of horizontal (e.g.…”