JournalsprimsVol. 39, No. 1pp. 117–164

Backward Global Solutions Characterizing Annihilation Dynamics of Travelling Fronts

  • Hiroki Yagisita

    Kyoto University, Japan
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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(xct) 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(xp_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