Global boundedness and the Allee effect in a nonlocal bistable reaction–diffusion equation in population dynamics

Abstract

This paper deals with the Cauchy problem of a nonlocal bistable reaction–diffusion equation ∂u∂t=Δu+μu2(1−κJ∗u)−γu,(x,t)∈RN×(0,∞)with N≤2, μ,κ,γ>0 and u(x,0)=u0(x). Under appropriate assumptions on J, it is proved that for any nonnegative and bounded initial condition, this problem admits a global bounded classical solution for N=1, while for N=2, global bounded classical solution exists for large κ values. Moreover, for small μ values and small initial data, the solution is shown to converge to 0 exponentially or locally uniformly as t→∞, which is referred as the Allee effect.