Global Solutions to a Nonlocal Fisher-KPP Type Problem
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Global Solutions to a Nonlocal Fisher-KPP Type Problem Shen Bian1
Received: 5 December 2015 / Accepted: 11 September 2016 © Springer Science+Business Media Dordrecht 2016
Abstract We consider a nonlocal Fisher-KPP reaction-diffusion model arising from popu lation dynamics, consisting of a certain type reaction term uα (1 − Ω uβ dx), where Ω is a bounded domain in Rn (n ≥ 1). The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of α, β. More precisely, for 1 ≤ α < 1 + (1 − 2/p)β, where p is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of n ≥ 3 and β = 1, α < 1 + 2/n is the known Fujita exponent (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966). Comparing to Fujita equation (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966), this paper will give an opposite result to our nonlocal problem. Keywords Fisher-KPP equation · Reaction-diffusion · Global existence
1 Introduction In this paper, we study the following nonlocal initial boundary value problem, ut − u = uα 1 − uβ (x, t)dx , x ∈ Ω, t > 0,
(1a)
Ω
∇u · ν = 0,
x ∈ ∂Ω,
(1b)
u(x, 0) = u0 (x) ≥ 0,
x ∈ Ω,
(1c)
where u is the density of population, Ω is a smooth bounded domain in Rn , n ≥ 1, α, β ≥ 1 and ν is the outer unit normal vector on ∂Ω. Without loss of generality, throughout this paper we assume |Ω| = 1 (otherwise, rescale the problem by |Ω|). Partially supported by National Science Foundation of China (Grant No. 11501025) and the Fundamental Research Funds for the Central Universities (Grant No. ZY1528).
B S. Bian
[email protected]
1
Beijing University of Chemical Technology, 100029, Beijing, China
S. Bian
This kind of model is developed to describe the population dynamics [6, 9] with the form ∂u ∂ 2 u = 2 + F (u), ∂t ∂x
(2)
2
∂ u where u is the population density, ∂x 2 describes the random displacement of the individuals of the population, the function F (u) is considered as the rate of the reproduction of the population. Its usual form is the local version
F (u) = uα (1 − u) − γ u,
(3)
the reaction term consists of the reproduction term which is represented by u to a power uα and (1 − u) which stands for the local consumption of available resources, the last term −γ u is the mortality of the population. The nonlocal version is ∞ ∂u ∂ 2 u α = 2 +u 1− φ(x − y)u(y, t)dy − γ u, ∂t ∂x −∞ ∞ where −∞ φ(y)dy = 1. φ(x − y) represents the probability density function that describes the distribution of individuals around their average positions. Noting that if φ is a Dirac δ function, the nonlocal problem reduces to the local version (3). In this paper, we will study the problem with nonlocal version reaction term. Nonlocal type reaction terms can describe also Darwinian evolution of a structured population density or the behaviors of cancer cells with therapy [5, 9]. There are some already known results on the r
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