Front propagation in periodic excitable media

By Henri Berestycki, François Hamel. 2002

433 citations recorded by the source corpus.

Subfields: Computer Networks and Communications, Control and Systems Engineering, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/d37770bd06596454b8a67c29f2633b28b8a1a585

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Citations in the corpus

Bistable traveling waves around an obstacle (2009, background): “Definitions and results about this case can be found in [3], [5], [6], [7], [19], [20] and [21].”
Traveling waves and spreading speeds for time-space periodic monotone systems (2015, background): “Hamel and Roques [14] gave a complete classification of all KPP pulsating fronts and obtained some global stability properties for the fronts including the one with minimal speed.”
Pulsating fronts for nonlocal dispersion and KPP nonlinearity (2013, positive): “To prove this result, we first work with an elliptic regularization Lκ of the operator M − Id + ε x − c∂s as it is done in [5,21,25] and introduce a truncated problem as follows.”
Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations (2012, background): “It is known that, for each direction e, there is a minimal speed c∗(e) > 0 such that pulsating travelling fronts Uc in the direction e exist if and only if c ≥ c∗(e), see [3, 41].”
Entire solutions in reaction-advection-diffusion equations in cylinders (2008, positive): “The entire solutions of (1.1) can be also viewed as orbits u(·, t), t ∈ R, lying in the space of the C2(Rnx) functions u ∈ (0,1), see Hamel and Nadirashvili [19].”
Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat (2014, background): “Many more existence results have actually been established for other types of nonlinearities f such as combustion, monostable or specific Fisher-KPP nonlinearities: for instance, for 1-periodic (in x) combustion nonlinearities f for which f ≥ 0 in R × [0, 1], f = 0 in R × ( [0, θ] ∪ {1} ) for some θ”
Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity (2008, methodology): “is a particular case of the result of [2, 3] which had been recalled above).”
Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait (2012, background): “Travelling front solutions in heterogeneous versions of (2) with periodicity in space, in time, or more general media are studied in [12], [28], [44], [6], [8], [36], [37].”
Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations (2015, positive): “Berestycki and Hamel showed in particular that the notion of spatial transition waves includes all the previous notions of fronts in homogeneous or periodic media, including non-trivial ones such as fronts with a speed which changes with respect to time (which have been proved to exist in [13]).”
Bistable Traveling Waves for Monotone Semiflows with Applications (2011, background): “For monostable systems in periodic habitats, we refer to [5, 8, 9, 22, 23, 26] and references therein.”
Uniqueness and stability properties of monostable pulsating fronts (2011, methodology): “Two given fronts could then automatically be compared globally in R × Ω, up to time-shifts, and a sliding method similar to [2] [3] would imply that the functions φ(s, x, y) are unique up to shifts in the variable s, and that the speed c, if any, is necessarily unique.”
Convergence to a single wave in the Fisher-KPP equation (2016, background): “Then there is a minimal speed c∗ > 0 such that for each c ≥ c∗, there is a unique pulsating front Uc(t, x), up to a time shift [4, 13].”
Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coefficients (2015, methodology): “7 part (iii) in [3] that the function k is convex.”
Traveling Waves in Discrete Periodic Media for Bistable Dynamics (2008, background): “[ 5 ], Berestycki et al. [6], Berestycki et al. [7,8], and Matano [34] for various heterogeneous media.”
Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems (2014, methodology): “[16] F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math.”
Front propagation into unstable states (2003, neutral): “The simple model problem which has often been studied in this context is the following extension of the F-KPP equation [47,48]”