Asymptotic spreading for general heterogeneous Fisher-KPP type equations / Henri Berestycki, Gregoire Nadin.

Format:

Book

Author/Creator:

Berestycki, H. (Henri), author.

Nadin, Gregoire, author.

Series:

Memoirs of the American Mathematical Society ; volume 280, number 1381.

Memoirs of the American Mathematical Society, 0065-9266 ; volume 280, number 1381

Language:

English

Subjects (All):

Reaction-diffusion equations.

Differential equations, Parabolic--Asymptotic theory.

Differential equations, Parabolic.

Physical Description:

vi, 100 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2022]

Summary:

"In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations. These are concerned with the dynamics of the solution starting from initial data with compact support. The nonlinearity f is of Fisher-KPP type, and admits 0 as an unstable steady state and 1 as a globally attractive one (or, more generally, admits entire solutions , where is unstable and is globally attractive). Here, the coefficients are only assumed to be uniformly elliptic, continuous and bounded in . To describe the spreading dynamics, we construct two non-empty star-shaped compact sets such that for all compact set (resp. all closed set , one has lim ). The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that and to establish an exact asymptotic speed of propagation in various frameworks. These include: almost periodic, asymptotically almost periodic, uniquely ergodic, slowly varying, radially periodic and random stationary ergodic equations. In dimension N, if the coefficients converge in radial segments, again we show that and this set is characterized using some geometric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets"-- Provided by publisher.

Contents:

A general formula for the expansion sets

Exact asymptotic spreading speed in different frameworks

Properties of the generalized principal eigenvalues

Proof of the spreading property

The homogeneous, periodic and compactly supported cases

The almost periodic case

The uniquely ergodic case

The radially periodic case

The space-independent case

The directionally homogeneous case

Proof of the spreading property with the alternative definition of the expansion sets and applications

Further examples and other open problems.

Notes:

Includes bibliographical references.

ISBN:

9781470454296

1470454297

OCLC:

1422208480