Uniqueness of critical traveling waves for nonlocal lattice equations with delays

Yu, Zhixian

doi:10.1090/s0002-9939-2012-11225-0

Cited by 23 publications

(15 citation statements)

References 15 publications

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“…To that aim, we argue by contradiction by assuming that, up to a subsequence, x n β → ∞. In that case, recalling (41), one obtains that U (x) 0 and U (x) β < 1, for all x ∈ R. This means that U (x) → θ as x → ∞, a contradiction with (39). Hence, {x n β } is bounded and there exists…”

Section: Proof Of Theorem 26 On Oscillationsmentioning

confidence: 94%

Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

Alfaro

Ducrot

Giletti

2017

Proc. London Math. Soc.

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We consider a bistable (0<θ<1 being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in +∞. Combining refined a priori estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in −∞ to ‘something’ which is strictly above the unstable equilibrium θ in +∞. Furthermore, we present situations (additional bound on the non‐linearity or small delay) where the wave converges to 1 in +∞, whereas the wave is shown to oscillate around 1 in +∞ when, typically, the delay is large.

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Section: Proof Of Theorem 26 On Oscillationsmentioning

confidence: 94%

Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

Alfaro

Ducrot

Giletti

2017

Proc. London Math. Soc.

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“…Finally, under weaker conditions on g, f , we get from Theorem 1.2 the following [1,8,15,16,20,23,24].…”

Section: Applicationsmentioning

confidence: 99%

Separation dichotomy and wavefronts for a nonlinear convolution equation

Gómez

Prado

Trofimchuk

2014

Journal of Mathematical Analysis and Applications

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This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation should either be asymptotically separated from zero or it should converge (exponentially) to zero. This dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi-wavefront solutions to the convolution equation. Finally, we apply our abstract results to several well-studied classes of evolution equations with asymmetric non-local and non-monotone response. We show that, contrary to the symmetric case, these equations can possess at the same time the stationary, the expansion and the extinction waves.Comment: 15 pages, submitte

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“…Some of such applications appear in the theory of the compound pendulum, surges in springs and connected systems of springs, equations of Born and von Kármán and elastic waves in a lattice. More recently, Hu and Li in [18] and Z. Yu in [40] examined the qualitative behaviour of the solutions of nonlocal semidiscrete equations with delay, which concerns the distribution of the mature population of a single species. Also, in [27] Mallet-Paret studied the existence of traveling waves for semidiscrete equations that include the well-known versions of the Nagumo and KPP-Fisher equations, highlighting their practical usefulness and main differences with the continuous case.…”

Section: Introductionmentioning

confidence: 99%

Well posedness for semidiscrete fractional Cauchy problems with finite delay

Lizama¹,

Murillo‐Arcila²

2018

Journal of Computational and Applied Mathematics

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We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delaywhere T is a bounded linear operator defined on a Banach space X (typically a space of functions like L p (Ω), 1 < p < ∞) and ∆ α corresponds to the time discretization of the continuous Riemann-Liouville fractional derivative by means of the Poisson distribution. We characterize the existence and uniqueness of solutions in vector-valued Lebesgue spaces of sequences of the model (0.1) in terms of boundedness of the operator-valued symbolwhenever 0 < α ≤ 1 and X satisfies a geometrical condition. For this purpose, we use methods from operator-valued Fourier multipliers and resolvent operator families associated to the homogeneous problem. We apply this result to show a practical and computational criterion in the context of Hilbert spaces.

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Uniqueness of critical traveling waves for nonlocal lattice equations with delays

Cited by 23 publications

References 15 publications

Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

Separation dichotomy and wavefronts for a nonlinear convolution equation

Well posedness for semidiscrete fractional Cauchy problems with finite delay

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