Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread

Abstract

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator 1 σ m Ω Jσ(x − y)(u(y, t) − u(x, t))dy and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalised principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter m ∈ [0, ∞). However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter m is in a different range. In particular, for the case m = 0, we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition. AMS Subject Classification (2010): 35R09; 45C05; 45M05; 45M20; 92D25.Since we only integrate over Ω, we assume that diffusion takes place only in Ω. The individuals may not enter or leave the domain, which is called nonlocal Neumann boundary condition, see Andreu-Vaillo et al. [1] and Cortázar et al. [9]. Throughout this paper, we will always make the following assumptions on the dispersal kernel J and the nonlinear function f :(J) J ∈ C(R N ) is nonnegative symmetric with compact support on the unit ball B 1 (0), J(0) > 0 and R N J(z)dz = 1.

The dynamics of nonlocal diffusion problems with a free boundary in heterogeneous environment

“…Proof. Since the proof method of (i) is similar to that of Theorem 2.1 in the work of Su et al [30], the details are not written here.…”

Age-structured Models with Nonlocal Diffusion of Dirichlet Type, I: Principal Spectral Theory and Limiting Properties

“…Note that the sets in Definition 5.1 are nonempty, see the proof of Theorem 5.3 in the following. As mentioned before, such ideas are widely used to prove the existence and asymptotic behavior of principal eigenvalues with respect to diffusion rate, see Coville [7], Li et al [30] and Su et al [49] for nonlocal diffusion equations, Shen and Vo [45] and Su et al [48] for time periodic nonlocal diffusion equations. As Shen and Vo [45] highlighted for the time periodic case, we remark that our parabolic-type operator A containing ∂ a is not self-adjoint, and thus we lack the usual L 2 (Ω) variational formula for the principal eigenvalue s(A).…”

“…We conjecture that the principal eigenvalue for age-structured models with nonlocal diffusion converges to the one for age-structured models with Laplace diffusion. Actually, without age-structure, the autonomous nonlocal diffusion operator has a L 2 variational structure which can be used to show the convergence, see Berestycki et al [3] and Su et al [49]. While for the time-periodic nonlocal diffusion operators, Shen and Xie [42,43] used the idea of a solution mapping to show the convergence, where they employed the spectral mapping theorem which is not valid in our case either, since we have a first order differential operator ∂ a that is unbounded.…”

“…Later Berestycki et al [3] further studied the problem in both bounded and unbounded domains, then investigated the asymptotic behavior of the generalized principal eigenvalue with respect to the diffusion rate. Related studies along this direction include Coville et al [8][9][10], Li et al [30], Su et al [49], Sun et al [51], Yang et al [56,57], Sun et al [50], Brasseur [5] and the references cited therein. On the other hand, Rawal and Shen [40], Rawal et al [41], Shen and Xie [42,43], and Shen and Zhang [44] investigated the existence of principal eigenvalues for autonomous and time periodic cases respectively, where they gave sufficient and necessary conditions for both cases by using the idea of perturbation of positive operators from B ürger [6].…”

The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

“…Nonlocal dispersal evolution equations of the form (1.2) have attracted a lot of attentions in recent years; See [29,31,33,36,40] and references therein. The case f (x, t, u) = f (x, u) in equations (1.2) has been well studied; See [3,5,7,9,12,20,34,35,[37][38][39]42]. We first recall the following results of the existence and non-existence of positive time-periodic solutions to (1.3) by Rawal and Shen [29] and Shen and Vo [36]: Lemma 1.6.…”