Threshold convergence results for a nonlocal time-delayed diffusion equation

Abstract

In this paper we consider the asymptotic behavior for nonlocal dispersion Nicholson blowflies equation ut=D(J⁎u−u)−du+pu(t−τ,x)e−au(t−τ,x) in the whole RN. By the method of Fourier transform, we first derive the decay estimates for the fundamental solutions with time-delay. Then, we obtain the threshold results with optimal convergence rates for the original solution to the constant equilibrium. Namely, when 0<pd<1, the solution u(t,x) globally converges to the equilibrium 0 in the time-exponential form; when pd=1, the solution u(t,x) globally converges to 0 in the time-algebraical form; when 1<pd≤e, the solution u(t,x) globally converges to u+ in the time-exponential form; and when e<pd<e2, it locally converges to u+ in the time-exponential form. This indicates that when the death rate is bigger than the birth rate, the blowflies will disappear in future. While, when the birth rate is bigger than the death rate in a certain range, then the blowflies population will reach an equilibrium after long time. The lower-higher frequency analysis plays a crucial role in the proof.