On uniqueness of semi-wavefronts

Aguerrea, Maitere; Gómez, Carlos M. Meléndez; Trofimchuk, Sergeĭ

doi:10.1007/s00208-011-0722-8

Cited by 63 publications

(86 citation statements)

References 33 publications

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“…Furthermore, when r ≥ r, if waves up to shift in this case has also been proved by the Diekmann-Kaper theory in [1]. Finally, when p d > e 2 , the wavefronts exist only for r in some bounded set [38,37], and the wavefronts are slowly oscillating at +∞ when the time-delay r satisfies (1.6).…”

Section: Introduction and Main Resultsmentioning

confidence: 67%

“…As shown in [35], So and Zou proved by the upper-lower solutions method that there exists a minimal wave speed c * = c * (r) > 0 (the so-called critical wave speed, which is given by the characteristic equation of the linearizing equation of (1.5) around the equilibrium v − = 0), when c ≥ c * , for any time-delay r > 0, the traveling wavefronts φ(x + ct) exist and are monotone. The uniqueness (up to a constant shift) was shown by Aguerrea, Gomez, and Trofimchuk [1] recently by means of the Diekmann-Kaper theory. However, when …”

Section: Introduction and Main Resultsmentioning

confidence: 71%

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Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Cheng

Lin

Lin³

et al. 2014

SIAM J. Math. Anal.

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Abstract. This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 < p d ≤ e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p d > e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e <

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Section: Introduction and Main Resultsmentioning

confidence: 67%

Section: Introduction and Main Resultsmentioning

confidence: 71%

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Cheng

Lin

Lin³

et al. 2014

SIAM J. Math. Anal.

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“…It can be monotone or oscillatory. The existence and uniqueness of the monotone/oscillatory traveling waves of (1.1) have been studied extensively [1,[3][4][5][6][7]14,29,32,33], see the references therein. We briefly describe the results we need below.…”

Section: Introductionmentioning

confidence: 99%

Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay

Chern

Mei

Yang

et al. 2015

Journal of Differential Equations

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This paper is concerned with the stability of critical traveling waves for a kind of non-monotone timedelayed reaction-diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction-diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves φ(x + ct) exist for all c ≥ c * and are exponentially stable for all wave speed c > c * [13], where c * is called the critical wave speed. In this paper, we prove that the critical traveling waves φ(x + c * t) (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results.

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“…With the semi-wave established above, we can construct various super-and subsolutions to estimate the spreading fronts h(t), g(t) and the spreading profile as t → ∞. 1 We sincerely thank Professor Avner Friedman for his valuable comments and suggestions on the proof of the well-posedness.…”

Section: Introductionmentioning

confidence: 99%

Propagation dynamics of Fisher–KPP equation with time delay and free boundaries

Sun

Fang

2019

Calc. Var.

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Incorporating free boundary into time-delayed reaction-diffusion equations yields a compatible condition that guarantees the well-posedness of the initial value problem. With the KPP type nonlinearity we then establish a vanishing-spreading dichotomy result. Further, when the spreading happens, we show that the spreading speed and spreading profile are nonlinearly determined by a delay-induced nonlocal semi-wave problem. It turns out that time delay slows down the spreading speed.2010 Mathematics Subject Classification. 35K57, 35R35, 35B40, 92D25.

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On uniqueness of semi-wavefronts

Cited by 63 publications

References 33 publications

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay

Propagation dynamics of Fisher–KPP equation with time delay and free boundaries

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