# Backward Global Solutions Characterizing Annihilation Dynamics of Travelling Fronts

### Hiroki Yagisita

Kyoto University, Japan

## Abstract

We consider a reaction-diffusion equation *u__t* = *u__xx* + *f*(*u*), where *f* has exactly three zeros 0, α and 1 (0 < α < 1), *f__u*(0) < 0, *f__u*(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 _u_0(*x*) with lim_x_→±∞ _u_0(*x*) > α having two interfaces at a large distance, *u*(*x, t*) approaches a pair of travelling wave solutions φ(*x* − *p_1( t)) + φ(−_x* + _p_2(

*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_−_x_0, _t_−_t_0).

## Cite this article

Hiroki Yagisita, Backward Global Solutions Characterizing Annihilation Dynamics of Travelling Fronts. Publ. Res. Inst. Math. Sci. 39 (2003), no. 1, pp. 117–164

DOI 10.2977/PRIMS/1145476150