Backward global solutions characterizing annihilation dynamics of travelling fronts (Q1873650)

The author analyses reaction-diffusion equation \(u_t = u_{xx} + f(u)\), where \(f\) has exactly three zeros at \(0 < \alpha < 1\), \(f_u (0) <0\), \(f_u (1) < 0\) and \(\int_0^1 f(u) du \geq 0\). Extending known results [\textit{P. C. Fife} and \textit{J. B. McLeod}, Arch. Ration. Mech. Anal. 65, 335-361 (1977; Zbl 0361.35035); \textit{D. Henry}, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Springer-Verlag (1981; Zbl 0456.35001)], he shows that the two travelling fronts \(\Psi (x-p_1(t))\) and \(\Psi (-x+p_2(t))\) approximating the solution for large \(t\) eventually disappear in mutual collision, and that the annihilation process is approximated by a solution \(\psi (x-x_0, t-t_0)\), whose properties are also investigated.