# 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 $∫_{0}f(u)du≥0$. 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