Abstract

This paper deals with the asymptotic behavior of solutions to the delayed monostable equation: (*) u(t)(t, x) = u(xx)(t, x)-u(t, x)+g(u(t-h, x)), x is an element of R, t > 0; here h > 0 and the reaction term g : R+ -> R+ is Lipschitz continuous and has exactly two fixed points (zero and kappa > 0). Under certain condition on the derivative of g at kappa (without assuming classic KPP condition for g) the global stability of fast semi-wavefronts is proved. Also, when the Lipschitz constant L-g is equal to g'(0) the stability of all semi-wavefronts (e.g., critical, non-critical and asymptotically periodic semi-wavefronts) on each interval in the form (-infinity, N], N is an element of R, to (*) is established, which includes classic equations such as the Nicholson's model.