Pulsating fronts for nonlocal dispersion and KPP nonlinearity

By Jérôme Coville, J. D'avila, Salomé Martínez. 2013

134 citations recorded by the source corpus.

Abstract In this paper we are interested in propagation phenomena for nonlocal reaction–diffusion equations of the type: ∂ u ∂ t = J ⁎ u − u + f ( x , u ) t ∈ R , x ∈ R N , where J is a probability density and f is a KPP nonlinearity periodic in the x variables. Under suitable assumptions we establish the existence of pulsating fronts describing the invasion of the 0 state by a heterogeneous state. We also give a variational characterization of the minimal speed of such pulsating fronts and exponential bounds on the asymptotic behavior of the solution.

Subfields: Modeling and Simulation, Public Health, Environmental and Occupational Health, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/37dee062d9ccaeeb53d0545976ed5e162d21d817

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Citations in the corpus

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): “Nonlocal versions of (2) where the Laplace operator is replaced by a nonlocal operator are studied in [19], [17, 18], [42].”
Traveling waves and spreading speeds for time-space periodic monotone systems (2015, background): “from a monotone dynamical system point of view to ﬁnd a link between the monostable subsystems and bistable system itself, which is used to establish the existence of bistable wavefronts. We refer to [21, 6, 34, 35] for nonlocal dispersal equations, and two survey papers [43, 15] for more references”
On eigenvalue problems arising from nonlocal diffusion models (2016, methodology): “In [10], for the case κ(x, y) = κ(y−x) it was observed that a positive and continuous eigenfunction φp exists if and only if λp < m.”
Propagation Phenomena for A Reaction–Advection–Diffusion Competition Model in A Periodic Habitat (2014, background): “[4] investigated spreading speeds and periodic traveling waves for a large class of such equations via quite different approaches.”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, methodology): “Moreover, as noticed in [25, 30, 41, 60], the operator L Ω + a may not have any eigenvalues in the space Lp(Ω) or C(Ω̄).”
Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats (2013, background): “In the case that f(t, x, u) = f(x, u) is spatially periodic, the existence of traveling waves with speed c = c∗(ξ) is also proved in [11] and the uniqueness and stability of traveling waves with speed c > c∗(ξ) are proved in [36].”