Exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries

Accelerated propagation is a new phenomenon associated with nonlocal diffusion problems. In this paper, we determine the exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries, where the nonlocal diffusion operator is given by ∫ RJ(x-y)u(t,y)dy-u(t,x) ∫ R J ( x - y ) u ( t , y ) d y - u ( t , x ) , and the kernel function J ( x ) behaves like a power function near infinity, namely lim |x|→ ∞ J(x)|x|α=λ >0 lim | x | → ∞ J ( x ) | x | α = λ > 0 for some α ∈ (1,2] α ∈ ( 1 , 2 ] . This is the precise range of α α where accelerated spreading can happen for such kernels. By constructing subtle upper and lower solutions, we prove that the location of the free boundaries $$x=h(t)$$ x = h ( t ) and $$x=g(t)$$ x = g ( t ) goes to infinity at exactly the following rates: $${aligned} {\{ {array}{ll} lim t→ ∞h(t)/tln t=lim t→ ∞-g (t)/tln t=μ λ ,&{} when α =2,\\ lim t→ ∞{h(t)}{ t1/(α -1)}= lim t→ ∞{-g (t)}{ t1/(α -1)}={22-α}{2-α }μ λ , &{} when α ∈ (1,2). {array}. } {aligned}$$ lim t → ∞ h ( t ) t ln t = lim t → ∞ - g ( t ) t ln t = μ λ , when α = 2