Bounds on the wave speed

Abstract EN

[1].

By constructing appropriate sub- and super-solutions to this system −cu′ = u′′ + f(u, v), −cv′ = ϵ2v′′ + g(u, v), (u, v)(−∞) = S−, (u, v)(∞) = S+ (1) Where, we are interested in component-wise monotone travelling wave solutions of the system of equations ut= uxx+ f(u, v), vt= ϵ2vxx+ g(u, v), (2) for (x, t ) ∈R × R+ for which the asymptotic conditions (u, v)(−∞, t) = S−, (u, v)(∞, t) = S+, t >0 (3) are satisfied.

Similar to those introduced in [3] and using essentially identical arguments, itcan be shown that −K ≤ c ≤ Lϵ,(4) whereK and L are positive constants independent of ϵ.

One immediate consequenceof this result is that in the limit ϵ → 0 only left travelling waves exist.We investigate the sharpness of these bounds in the special case of CLV kinetics.We show that: the bounds of the wave speed given in [4] are optimal for the given leftand right solutions (sub-solutions and super-solutions).