Pulsating type entire solutions of monostable reaction–advection–diffusion equations in periodic excitable media

Liu, Nai-Wei; Li, Wan–Tong; Wang, Zhicheng

doi:10.1016/j.na.2011.09.037

Cited by 12 publications

(4 citation statements)

References 42 publications

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“…Note that all these works mainly concerned with space/time homogeneous equations. Recently, many researchers devoted to the study of entire solutions for space/time periodic equations, see, e.g., [6,28,30]. In particular, Du et al [10] established the invasion entire solutions for system (1.1) with monostable structure.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system

Wang

2018

Journal of Differential Equations

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This paper is concerned with a time periodic competition-diffusion systemwhere u(t, x) and v(t, x) denote the densities of two competing species, d > 0 is some constant, r i (t), a i (t) and b i (t) are T −periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang [J. Differential Equations 255 (2013), 2402-2435] that this system admits a periodic traveling front connecting two stable semi-trivial T −periodic solutions (p(t), 0) and (0, q(t)) associated to the corresponding kinetic system. Assume further that the wave speed is non-zero, we investigate the asymptotic behavior of the periodic bistable traveling front at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then give some key estimates. Finally, by applying super-and subsolutions technique as well as the comparison principle, we establish the existence and various qualitative properties of entire solutions defined for all time and whole space.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system

Wang

2018

Journal of Differential Equations

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“…Over the last couple of decades, the existence, uniqueness, qualitative properties, and stability properties of solutions have been extensively studied for nonlinear advection-reaction-diffusion equations, see [3,6,8,13,15,17,19,20,22,23]. In the special cases where F does not possess advection term, under several possible assumptions on the nonlinearity, the existence, uniqueness, stability properties of the special solutions and influence on the dynamics of the problems have been investigated in [19,20,22].…”

Section: Introductionmentioning

confidence: 99%

Iterative solution for nonlinear impulsive advection- reaction-diffusion equations

Hao¹,

Liu²,

Wu³

2016

J. Nonlinear Sci. Appl.

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Through solving equations step by step and by using the generalized Banach fixed point theorem, under simple conditions, the authors present the existence and uniqueness theorem of the iterative solution for nonlinear advection-reaction-diffusion equations with impulsive effects. An explicit iterative scheme for the solution is also derived. The results obtained generalize and improve some known results.

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“…For high dimensional spaces, Li et al [19] considered the existence of entire solutions of a reactionadvection-diffusion equation with monostable and ignition temperature nonlinearities in infinite-cylinders by considering a pair of traveling curved fronts, and Liu et al [22] considered the similar problem for bistable nonlinearity. See also [16] for an amazingly rich class of entire solutions by using planar traveling wave fronts and [21] for periodic media.…”

Section: Introductionmentioning

confidence: 99%

Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media

Liu

2010

Sci. China Math.

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This paper is concerned with entire solutions (t ∈ R) for bistable reaction-advection-diffusion equations in heterogeneous media. By using traveling curved fronts connecting a constant unstable stationary state and a stable stationary state, we proved that there exist entire solutions behaving as two traveling curved fronts coming from opposite directions, and approaching each other. Furthermore, we prove that such an entire solution is unique and Liapunov stable. The key technique is to characterize the asymptotic behavior of solutions at infinity in term of appropriate subsolutions and supersolutions.

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Pulsating type entire solutions of monostable reaction–advection–diffusion equations in periodic excitable media

Cited by 12 publications

References 42 publications

Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system

Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system

Iterative solution for nonlinear impulsive advection- reaction-diffusion equations

Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media

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