Asymptotic behaviour of solutions of free boundary problems for Fisher-KPP equation

We study a free boundary problem for the Fisher-KPP equation: u(t) = u(xx) + f(u) (g(t) < x < h(t)) with free boundary conditions h' (t) = -u(x)(t, h(t)) - alpha and g' (t) = -u(x)(t, g(t)) + beta for 0 < beta < alpha. This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We investigate the asymptotic behaviour of bounded solutions. There are two parameters alpha(0) and alpha* with 0 < alpha(0) < alpha* which play key roles in the dynamics. More precisely, (i) in case 0 < beta < alpha(0) and 0 < alpha < alpha*, we obtain a trichotomy result: (i-1) spreading, i.e., h(t) - g(t) -> + infinity and u(t, . + ct) -> 1 with c epsilon (c(L), c(R)), where c(L) and c(R) are the asymptotic spreading speed of g(t) and h(t), respectively, (c(R) > 0 > c(L) when 0 < beta < alpha < alpha(0); c(R) = 0 > c(L) when 0 < beta < alpha = alpha(0); 0 > c(R) > c(L) when alpha(0) < alpha < alpha* and 0 < beta < alpha(0)); (i-2) vanishing, i.e., lim(t -> T) h(t) = lim(t -> T) g(t) and lim(t -> T) u(t, x) = 0, where T is some positive constant; (i-3) transition, i.e., g(t) -> -infinity, h(t) -> -infinity, 0 < lim(t -> 8)[h(t) - g(t)] < + infinity and u(t, x). V*(x - c* t) with c* < 0, where V*(x - c* t) is a travelling wave with compact support and which satisfies the free boundary conditions. (ii) in case beta >= alpha(0) or alpha >= alpha*, vanishing happens for any solution.