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

Abstract

AbstractAccelerated 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 $$\displaystyle \int _{\mathbb {R}}J(x-y)u(t,y)dy-u(t,x)$$ ∫ 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 $$\lim _{|x|\rightarrow \infty } J(x)|x|^{\alpha }=\lambda >0$$ lim | x | → ∞ J ( x ) | x | α = λ > 0 for some $$\alpha \in (1,2]$$ α ∈ ( 1 , 2 ] . This is the precise range of $$\alpha $$ 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: $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \lim _{t\rightarrow \infty }\frac{h(t)}{t\ln t}=\lim _{t\rightarrow \infty }\frac{-g (t)}{t\ln t}=\mu \lambda ,&{} \hbox { when } \alpha =2,\\ \displaystyle \lim _{t\rightarrow \infty }\frac{h(t)}{ t^{1/(\alpha -1)}}= \lim _{t\rightarrow \infty }\frac{-g (t)}{ t^{1/(\alpha -1)}}=\frac{2^{2-\alpha }}{2-\alpha }\mu \lambda , &{} \hbox { when } \alpha \in (1,2). \end{array}\right. } \end{aligned}$$ lim t → ∞ h ( t ) t ln t = lim t → ∞ - g ( t ) t ln t = μ λ , when α = 2 , lim t → ∞ h ( t ) t 1 / ( α - 1 ) = lim t → ∞ - g ( t ) t 1 / ( α - 1 ) = 2 2 - α 2 - α μ λ , when α ∈ ( 1 , 2 ) . Here $$\mu >0$$ μ > 0 is a given parameter in the free boundary condition. Accelerated propagation can also happen when $$\lim _{|x|\rightarrow \infty }J(x)|x|(\ln |x|)^\beta =\lambda >0$$ lim | x | → ∞ J ( x ) | x | ( ln | x | ) β = λ > 0 for some $$\beta >1$$ β > 1 . For this case, we prove that $$\begin{aligned} -g (t), h(t)=\exp \Big \{\Big [\Big (\frac{2\beta \mu \lambda }{\beta -1}\Big )^{1/\beta }+o(1)\Big ]t^{1/\beta }\Big \} \hbox { as } t\rightarrow \infty . \end{aligned}$$ - g ( t ) , h ( t ) = exp { [ ( 2 β μ λ β - 1 ) 1 / β + o ( 1 ) ] t 1 / β } as t → ∞ . These results considerably sharpen the corresponding ones in [20], and the techniques developed here open the door for obtaining similar precise results for other problems. A crucial technical point is that such precise conclusions on the propagation are achievable by finding the correct improvements on the form of the lower solutions used in [20], even though the precise long-time profile of the density function u(t, x) is still lacking.