The Influence of Fractional Diffusion in Fisher-KPP Equations

By X. Cabré, Jean-Michel Roquejoffre. 2012

187 citations recorded by the source corpus.

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/467a762fb3735bdb8d240cc47b00aa6eae549d69

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Modeling and Simulation

In Public Health, Environmental and Occupational Health

Citations in the corpus

The Fisher–KPP problem with doubly nonlinear “fast” diffusion (2016, methodology): “Even though this idea was firstly used in [16], we stress that our construction is completely independent by the previous one, and the comparison is not done in the whole space, but only on a sub-region.”
The Fisher-KPP problem with doubly nonlinear diffusion (2016, methodology): “Exponential propagation happens also with fractional diffusion, both linear and nonlinear, see [10] and [48] and their references.”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, background): “On the other hand, and independently, Cabré and Roquejoffre ([33]) studied the case of fractional linear diffusion ut(x, t)+(−∆)u(x, t) = f(u), where (−∆) is the Fractional Laplacian operator with s ∈ (0, 1) and concluded in the same vein that there is no traveling wave behavior as t → ∞, and indeed”