Backward global solutions characterizing annihilation dynamics of travelling fronts

Abstract

We consider a reaction-diffusion equation ut = uxx + f(u), where f has exactly three zeros 0, α and 1 (0 < α < 1), fu(0) < 0, fu(1) < 0 and ∫01 f(u)du ≥ 0. Then, the equation has a travelling wave solution u(x, t) = φ(x - ct) with φ(-∞) = 0 and φ(+∞) = 1. Known results suggest that for an initial state u0(x) with limx→±∞u0(x) > α having two interfaces at a large distance, u(x, t) approaches a pair of travelling wave solutions φ(x - p1(t)) + φ(-x + p2(t)) for a long time, and then the travelling fronts eventually disappear by colliding with each other. While our results establish this process, they show that there is a (backward) global solution ψ (x, t) and that the annihilation process is approximated by a solution ψ (x-x0, t-t0).