Nonlocal anisotropic dispersal with monostable nonlinearity

By Jérôme Coville, J. D'avila, Salomé Martínez. 2008

171 citations recorded by the source corpus.

Abstract We study the travelling wave problem J ⋆ u − u − c u ′ + f ( u ) = 0 in R , u ( − ∞ ) = 0 , u ( + ∞ ) = 1 with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for c ≠ 0 . For c = 0 we show examples of nonuniqueness.

Subfields: Applied Mathematics, Mathematical Physics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/75c96db66f70feef8ca15d6eb8a25e5c134f82bf

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In Applied Mathematics

In Mathematical Physics

In Public Health, Environmental and Occupational Health

Citations in the corpus

On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators (2010, background): “Such type of solution is commonly used to analyse the long-time behaviour of some nonlocal evolution problems [10, 14] and had proven to be a very efficient tool in the analysis of nonlinear integrodifferential problems; see for example [21, 31].”
Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity (2012, background): “3 in [18] shows that there exists a nondecreasing solution (φ, c) of (1.”
Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats (2010, unknown): “See, for example, [15], [16], [18], [54], [55] for the study of the existence of spreading speeds and traveling wave solutions connecting the trivial solution u = 0 and a nontrivial positive stationary solution.”
On uniqueness of semi-wavefronts (2010, methodology): “For instance, as it was established recently by Coville, Dávila and Mart́ınez [8], neither (3) nor g′(s) ≤ g′(0), s ∈ (0, 1), is necessary to prove the uniqueness of non-stationary monotone traveling fronts to (2).”
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 Wave Solutions of Spatially Periodic Nonlocal Monostable Equations (2012, background): “See, for example, [11], [13], [15], [36], [41], [45], [57], [58], for the study of entire solutions and the existence of spreading speeds and traveling wave solutions connecting the trivial solution u = 0 and a nontrivial positive stationary solution for some special cases of (1.”
Pulsating fronts for nonlocal dispersion and KPP nonlinearity (2013, unknown): “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.”