On the bounded solutions of a nonlinear convolution equation

Diekmann, Odo; Kaper, Hans G.

doi:10.1016/0362-546x(78)90015-9

Cited by 134 publications

(129 citation statements)

References 9 publications

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“…Similar methods were used to establish existence of travelling waves of some other nonlocal models with monostable dynamics; see, e.g., [1], [4], [6], [13], [11], [14], [5].…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness of travelling waves for nonlocal monostable equations

Carr

Chmaj

2004

Proc. Amer. Math. Soc.

293

246

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Abstract. We consider a nonlocal analogue of the Fisher-KPP equationand its discrete counterpartun = (J * u)n − un + f (un), n ∈ Z, and show that travelling wave solutions of these equations that are bounded between 0 and 1 are unique up to translation. Our proof requires finding exact a priori asymptotics of a travelling wave. This we accomplish with the help of Ikehara's Theorem (which is a Tauberian theorem for Laplace transforms).

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“…Similar methods were used to establish existence of travelling waves of some other nonlocal models with monostable dynamics; see, e.g., [1], [4], [6], [13], [11], [14], [5].…”

Section: Introductionmentioning

confidence: 99%

“…The first uniqueness result, for a particular monostable nonlocal model, appeared in the work of Diekmann and Kaper [6]. The authors proved that travelling waves with noncritical speeds (i.e., c > c 0 ) are unique in the class of monotone solutions.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of travelling waves for nonlocal monostable equations

Carr

Chmaj

2004

Proc. Amer. Math. Soc.

293

246

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“…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]). …”

Section: A Priori Estimate Of Traveling Wave Frontsmentioning

confidence: 99%

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

Wang

Ruan

2013

Math. Model. Nat. Phenom.

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Abstract. This paper is concerned with entire solutions of a class of bistable delayed lattice differential equations with nonlocal interaction. Here an entire solution is meant by a solution defined for all (n, t) ∈ Z × R. Assuming that the equation has an increasing traveling wave front with nonzero wave speed and using a comparison argument, we obtain a two-dimensional manifold of entire solutions. In particular, it is shown that the traveling wave fronts are on the boundary of the manifold. Furthermore, uniqueness and stability of such entire solutions are studied.

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“…Our method is similar to that of Carr and Chmaj [6] which has been used by Wang et al [43] (see also Diekmann and Kaper [11]). We first provide a technical lemma about the asymptotic behavior of a positive decreasing function, which is given by Carr and Chmaj [6,Proposition 2.3] and is important to prove our results.…”

Section: Lemma 32mentioning

confidence: 99%

“…Since U (t) > 0, there exists a real number ϑ such that (λ) is analytic for ϑ < Reλ < 0 and (λ) has a singularity at λ = ϑ. Hence, (λ) is defined for Reλ > λ 11 . We rewrite (3.4) as…”

Section: Theorem 35 Assume That φ (T) Is An Increasing Traveling Wamentioning

confidence: 99%

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

Ruan

2008

Trans. Amer. Math. Soc.

115

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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.

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On the bounded solutions of a nonlinear convolution equation

Cited by 134 publications

References 9 publications

Uniqueness of travelling waves for nonlocal monostable equations

Uniqueness of travelling waves for nonlocal monostable equations

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

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

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