The speed of propagation for KPP type problems. I: Periodic framework

By Henri Berestycki, François Hamel, Nikolai Nadirashvili. 2005

324 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/faf2a76f933995a9cca75b10b644b93956111a66

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In Numerical Analysis

In Public Health, Environmental and Occupational Health

Citations in the corpus

Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat (2014, methodology): “Furthermore, some variational formulas for cL in the positive case and in the more specific KPP case have been derived in [10, 12, 20] and some further qualitative and uniqueness properties for each speed c ∈ [cL,+∞) have been given in [25, 28, 29].”
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.”
Spreading speeds for one-dimensional monostable reaction-diffusion equations (2016, background): “This speed is also known to be the minimal speed of pulsating travelling waves [5].”
Asymptotic spreading in heterogeneous diffusive excitable media (2008, background): “Neither is it optimal in space periodic media, as it has been observed in [7]: there exist some speeds c > 2 √ γ infx∈RN f ′ u(x, 0) such that lim inft→+∞ sup|x|≤ct u(t, x) > 0, even for KPP nonlinearities.”
Analysis of the periodically fragmented environment model: II—biological invasions and pulsating travelling fronts (2005, background): “A formula for the minimal speed is given in [ 7 ] under the assumption thatf(x,s) fu(x,0)s for alls 2 [0,1] and the dependence ofc in terms of the diusion, advection, reaction coecients as well as the geometry of the domain, is analyzed.”
Uniqueness and stability properties of monostable pulsating fronts (2011, unknown): “…L λ ψ := −∇ · (A∇ψ) + 2λeA∇ψ + q · ∇ψ + [λ∇ · (Ae) − λq · e − λ 2 eAe − ζ − ]ψ (1.12) acting on the set of C 2 (Ω) periodic functions ψ such that νA∇ψ = λ(νAe)ψ on ∂Ω (see [5], actually, this existence result has been proved under additional smoothness assumptions on the coefficients of (1.1)).”
Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena (2005, background): “The right-hand side of (4.14) only depends on e, the domain � , the coefficients Av and a, and the dependence on f is only through the f � u (·, ·, 0). Let us also mention that the equality holds in (4.14) under the additional assumption f (x, y, u) ≤ f � u(x, y, 0)u for all u ∈ [0, 1] and for all (”
Propagation dynamics for monotone evolution systems without spatial translation invariance (2020, methodology): “ing habitat. 6 Applications In this section, we apply the results obtained in Sections 4 and 5 to four classes of monotone evolution equations. We start with the deﬁnition of KPP property (see, e.g., [5, 6]). Deﬁnition 6.1. Let u∗ ∈ (0,∞) and F : R+ → R+ be a continuously diﬀerentiable function. We ”
Front propagation for discrete periodic monostable equations (2006, methodology): “For more recent works, we refer the readers to [1, 2 , 3, 4, 24] and the references cited therein.”