Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity

Abstract

Abstract. This paper is concerned with entire solutions for bistable reactiondiffusion equations with nonlocal delay in one-dimensional spatial domain. Here the entire solutions are defined in the whole space and for all time t ∈ R. Assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument, we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x-axis and annihilating at a finite time. Furthermore, we show that such an entire solution is unique up to space-time translations and is Liapunov stable. A key idea is to characterize the asymptotic behavior of the solutions as t → −∞ in terms of appropriate subsolutions and supersolutions. In order to illustrate our main results, two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are considered.

Travelling waves of a diffusive Kermack–McKendrick epidemic model with non-local delayed transmission

“…Our construction of the invariant set is motivated by the work of . For c ∈ [0, c * ), we conclude the non-existence of non-trivial travelling wave solutions by an argument applying the Laplace transform to the I (x + ct) component, this argument was first introduced by Carr and Chmaj (2004) and further used by Wang et al (2008Wang et al ( , 2009.…”

Entire Solutions in Lattice Delayed Differential Equations with Nonlocal Interaction: Bistable Cases

“…Under the assumptions (H1)-(H3), it follows from Ma and Wu [32] and Wang et al [43] that (1.2) admits a strictly increasing traveling wave front ϕ (x + ct) satisfying ϕ (−∞) = 0 and ϕ (+∞) = K with wave speed c ∈ R. When c = 0, we know from [44] that for any (θ 1 , θ 2 ) ∈ R 2 , there exists a unique entire solution U (x, t) := U (x, t; θ 1 , θ 2 ) with 0 < U (x, t) < K for all (x, t) ∈ R 2 such that…”

“…For the delayed reaction-diffusion equation (1.2), we [44] established the existence and uniqueness of entire solutions behaving as two traveling fronts coming from opposite directions and approaching each other except for…”

“…Our method is similar to that of Carr and Chmaj [10] which has been used by Wang et al [44] (see also Diekmann and Kaper [21]). …”

“…For more details, we refer to Berestycki and Hamel [3,4] and the references therein. Recently, Li et al [30] and Wang et al [44] considered entire solutions of nonlocal reactiondiffusion equations with delayed nonlinearity (which covers (1.2)) for the monostable and bistable cases, respectively. For equation (1.1), Wang et al [45] studied the existence and uniqueness of its entire solutions under the monostable assumption.…”