Entire Solutions for Nonlocal Dispersal Equations with Bistable Nonlinearity: Asymmetric Case

Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity

“…Remark 1. In this paper, we have considered the existence, uniqueness, asymptotic behavior and Lyapunov stability of entire solutions of the nonlocal dispersal equation 1 under the assumption J(x) = J(−x), x ∈ R. It is well-known from [8,22,28,29] that the asymmetry of J has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. Naturally, an interesting problem is to consider entire solutions of 1 under asymmetric conditions.…”

Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel

“…To obtain the precise exponential asymptotic behaviors of φ(ξ) as ξ → +∞ andφ(ξ) asξ → −∞ (described as (c) of (i) and (ii) in Theorem 3.1), we use a similar argument with Lemma 3.4 in [22]…”

“…(ii) The equation F 2 (µ) = 0 (orF 2 (μ) = 0) has two real roots µ 21 < 0 (orμ 21 < 0) and µ 22 > 0 (orμ 22 > 0) such that From (2.6), we have…”

“…In [16,22], we have investigated the entire solutions of the asymmetric dispersal equation (1.1) with monostable and bistable nonlinearities, respectively. Compared with the symmetric case [9,14], we found that it occurs new types of entire solutions due to the influence of the asymmetry of the kernel function.…”

Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread

“…A typical example of such a nonlinearity is given by f (x, s) := s(a(x) − s) with a ∈ C(Ω). Nonlocal dispersal equations of the form (1.1) are widely used to model dispersal phenomena which exhibit nonlocal internal interactions and have attracted much attention, see Bates and Zhao [2], Berestycki et al [4], Cao et al [7], Chasseigne et al [8], Cortázar et al [9], Coville [10,13], Coville et al [11], Fife [16], Hutson et al [18], Kao et al [19], Li et al [22][23][24][25], Shen and Xie [28], Shen and Zhang [30], Su et al [32], Sun et al [35,36], Wang and Lv [37], Yang et al [38] and Zhang et al [39][40][41]. From both mathematical and ecological points of view, nonlocal dispersal kernel functions can have a variety of forms.…”