Backward Global Solutions Characterizing Annihilation Dynamics of Travelling Fronts

Abstract

We consider a reaction-diffusion equation $u_t=u_{xx}+f(u)$, where $f$ has exactly three zeros $0$, $\alpha$ and $1$ $(0<\alpha <1)$, $f_u(0)<0$, $f_u(1)<0$ and ${\int }_0^{1}f(u)du\ge 0$. Then, the equation has a travelling wave solution $u(x,t)=\varphi (x-ct)$ with $\varphi (-\infty )=0$ and $\varphi (+\infty )=1$. Known results suggest that for an initial state $u_0(x)$ with ${\underline{\lim }}_{x\to \pm \infty }{u}_0(x)>\alpha$ having two interfaces at a large distance, $u(x,t)$ approaches a pair of travelling wave solutions $\varphi (x-p_1(t))+\varphi (-x+p_2(t))$ 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 $\psi (x,t)$ and that the annihilation process is approximated by a solution $\psi (x-x_0,t-t_0)$.