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Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations

About this Title

Henri Berestycki and Grégoire Nadin

Publication: Memoirs of the American Mathematical Society
Publication Year: 2022; Volume 280, Number 1381
ISBNs: 978-1-4704-5429-6 (print); 978-1-4704-7281-8 (online)
DOI: https://doi.org/10.1090/memo/1381
Published electronically: October 7, 2022
Keywords: Reaction-diffusion equations, heterogeneous reaction-diffusion equations, propagation and spreading properties, principal eigenvalues, linear parabolic operator, Hamilton-Jacobi equations, homogenization, almost periodicity, unique ergodicity, slowly oscillating media

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Table of Contents

Chapters

• 1. Introduction
• 2. A general formula for the expansion sets
• 3. Exact asymptotic spreading speed in different frameworks
• 4. Properties of the generalized principal eigenvalues
• 5. Proof of the spreading property
• 6. The homogeneous, periodic and compactly supported cases
• 7. The almost periodic case
• 8. The uniquely ergodic case
• 9. The radially periodic case
• 10. The space-independent case
• 11. The directionally homogeneous case
• 12. Proof of the spreading property with the alternative definition of the expansion sets and applications
• 13. Further examples and other open problems

Abstract

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: \begin{equation*} \partial _{t} u - \sum _{i,j=1}^N a_{i,j}(t,x)\partial _{ij}u-\sum _{i=1}^N q_i(t,x)\partial _i u=f(t,x,u). \end{equation*} 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 $p^\pm (t,x)$, where $p^-$ is unstable and $p^+$ is globally attractive). Here, the coefficients $a_{i,j}, q_i, f$ are only assumed to be uniformly elliptic, continuous and bounded in $(t,x)$. To describe the spreading dynamics, we construct two non-empty star-shaped compact sets $\underline {\mathcal {S}}\subset \overline {\mathcal {S}} \subset \mathbb {R}^N$ such that for all compact set $K\subset \mathrm {int}(\underline {\mathcal {S}})$ (resp. all closed set $F\subset \mathbb {R}^N\backslash \overline {\mathcal {S}}$), one has $\lim _{t\to +\infty } \sup _{x\in tK} |u(t,x)-1| = 0$ (resp. $\lim _{t\to +\infty } \sup _{x\in tF} |u(t,x)| =0$).

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 $\overline {\mathcal {S}}=\underline {\mathcal {S}}$ 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 $\overline {\mathcal {S}}=\underline {\mathcal {S}}$ and this set is characterized using some geometric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets.