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Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments II: Flux-limited solutions

King-Yeung Lam, Gregoire Nadin, Xiao Yu

Abstract

We consider the spreading dynamics of the Fisher-KPP equation in a shifting environment, by analyzing the limit of the rate function of the solutions. For environments with a weak monotone condition, it was demonstrated in a previous paper that the rate function converges to the unique Ishii solution of the underlying Hamilton-Jacobi equations. In case the environment does not satisfy the weak monotone condition, we show that the rate function is then characterized by the Hamilton-Jacobi equation with a dynamic junction condition, which depends additionally on the generalized eigenvalue derived from the environmental function. Our results applies to the case when the environment has multiple shifting speeds, and clarify the connection with previous results on nonlocally pulled fronts and forced traveling waves.

Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments II: Flux-limited solutions

Abstract

We consider the spreading dynamics of the Fisher-KPP equation in a shifting environment, by analyzing the limit of the rate function of the solutions. For environments with a weak monotone condition, it was demonstrated in a previous paper that the rate function converges to the unique Ishii solution of the underlying Hamilton-Jacobi equations. In case the environment does not satisfy the weak monotone condition, we show that the rate function is then characterized by the Hamilton-Jacobi equation with a dynamic junction condition, which depends additionally on the generalized eigenvalue derived from the environmental function. Our results applies to the case when the environment has multiple shifting speeds, and clarify the connection with previous results on nonlocally pulled fronts and forced traveling waves.

Paper Structure

This paper contains 22 sections, 30 theorems, 220 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Statements of Results
  3. Flux-limited solution due to Imbert and Monneau
  4. Main results
  5. Monotonicity of the flux-limited solutions
  6. Generalizations
  7. Related optimal control formulations
  8. An earlier result: Viscosity solution in the sense of Ishii
  9. Applications and Explicit Formulas
  10. Preliminary Results
  11. The eigenvalue problem associated with $g(y)$
  12. The continuity of subsolutions
  13. Critical slope lemmas
  14. Proof of Main Results
  15. Verification of flux-limited solutions property
  16. ...and 7 more sections

Key Result

Proposition 2.3

Let $A \in \mathbb{R}$ be given. If $\underline\rho$ and $\overline\rho$ are, respectively, the FL-subsolution and FL-supersolution of eq:fl, and such that then $\underline\rho(s) \leq \overline\rho(s)$ in $[0,+\infty)$.

Figures (1)

  • Figure 1: The dependence of spreading speed $c_*$ on $c_1$. Here $r_\pm = g(\pm \infty)$. The case $c_*= c_1$ is also indicated in Berestycki2018forced. Nonlocal pulling is illustrated by the curved part of the blue lines in panels (a), (b), (d) and (f). The part where $c_*$ coincides either with the KPP speed of the limiting system at $\pm\infty$ is indicated by the horizontal part of the blue lines in (a)-(f).

Theorems & Definitions (76)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Remark 2.5
  • Theorem 1
  • Remark 2.6
  • Remark 2.7
  • Corollary 2.8
  • proof
  • ...and 66 more