Nonplanar traveling fronts in reaction–diffusion equations with combustion and degenerate Fisher-KPP nonlinearities

Wang, Zhicheng; Bu, Zhen-Hui

doi:10.1016/j.jde.2015.12.045

Cited by 44 publications

(16 citation statements)

References 36 publications

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“…Secondly, we remark that there also are many studies on the nonplanar traveling wave solutions of the scalar reaction-diffusion equations, for example, see [5,6,7,8,9,10,11,14,15,16,17,18,19,20,23,30,31,34,32,33,35,36,37,38,39,42,43,46].…”

Section: Xiongxiong Bao Wan-tong LI and Zhi-cheng Wangmentioning

confidence: 99%

Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system

Bao¹,

Li²,

Wang³

2020

Communications on Pure &Amp; Applied Analysis

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The existence, non-existence and qualitative properties of time periodic pyramidal traveling front solutions for the time periodic Lotka-Volterra competition-diffusion system have already been studied in R N with N ≥ 3. In this paper, we continue to study the uniqueness and asymptotic stability of such time-periodic pyramidal traveling front in the three-dimensional whole space. For any given admissible pyramid, we show that the time periodic pyramidal traveling front is uniquely determined and it is asymptotically stable under the condition that given perturbations decay at infinity. Moreover, the time periodic pyramidal traveling front is uniquely determined as a combination of two-dimensional periodic V-form waves on the edges of the pyramid.

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Section: Xiongxiong Bao Wan-tong LI and Zhi-cheng Wangmentioning

confidence: 99%

Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system

Bao¹,

Li²,

Wang³

2020

Communications on Pure &Amp; Applied Analysis

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“…In another paper, by using comparison principle, Sheng [12] studied the existence and stability of time-periodic traveling curved fronts about bistable reaction-diffusion equations in R 3 . In [13], Wang and Bu considered traveling curved fronts (nonplanar) for combustion and degenerate Fisher-KPP type reaction-diffusion equations. Ninomiya and Taniguchi [9,10] and Taniguchi [14,15] showed the existence and the stability of traveling curved fronts for Allen-Cahn equations.…”

Section: Introductionmentioning

confidence: 99%

Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations in R2

Liu

2017

Journal of Mathematics

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We consider the interaction of traveling curved fronts in bistable reaction-diffusion equations in two-dimensional spaces. We first characterize the growth of the traveling curved fronts at infinity; then by constructing appropriate subsolutions and supersolutions, we prove that the solution of the Cauchy problem converges to a pair of diverging traveling curved fronts in R2 under appropriate initial conditions.

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“…In fact, it follows from [1,10] that the assumptions (H1)-(H2) hold with c * f being the minimal wave speed and the unique wave speed of planar traveling front φ f when the nonlinear reaction term f is of degenerate Fisher-KPP monostable type and combustion type, respectively. See [2,19] for more details.…”

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confidence: 99%

Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations

2021

era

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<p style='text-indent:20px;'>In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted <inline-formula><tex-math id="M1">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula> spaces on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{n}\; (n\geq4) $\end{document}</tex-math></inline-formula>. Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^{4} $\end{document}</tex-math></inline-formula>.</p>

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Nonplanar traveling fronts in reaction–diffusion equations with combustion and degenerate Fisher-KPP nonlinearities

Cited by 44 publications

References 36 publications

Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system

Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system

Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations in R2

Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations

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