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The Bramson correction in the Fisher-KPP equation: from delay to advance

Matthieu Alfaro, Thomas Giletti, Dongyuan Xiao

Abstract

We consider the solution to the scalar Fisher-KPP equation with front-like initial data, focusing on the location of its level sets at large times, particularly their deviation from points moving at the known spreading speed. We consider an intermediate case for the tail of the initial data, where the decay rate approaches, up to a polynomial term, that of the traveling wave with minimal speed. This approach enables us to capture deviations of the form $-r \ln t$ with $r \< \frac{3}{2}$, which corresponds to a logarithmic delay when $0 \< r \< \frac{3}{2}$ and a logarithmic advance when $r \< 0$. The critical case $r=\frac 32$ is also studied, revealing an extra $\mathcal O(\ln \ln t)$ term. Our arguments involve the construction of new sub- and super-solutions based on preliminary formal computations on the equation with a moving Dirichlet condition. Finally, convergence to the traveling wave with minimal speed is addressed.

The Bramson correction in the Fisher-KPP equation: from delay to advance

Abstract

We consider the solution to the scalar Fisher-KPP equation with front-like initial data, focusing on the location of its level sets at large times, particularly their deviation from points moving at the known spreading speed. We consider an intermediate case for the tail of the initial data, where the decay rate approaches, up to a polynomial term, that of the traveling wave with minimal speed. This approach enables us to capture deviations of the form with , which corresponds to a logarithmic delay when and a logarithmic advance when . The critical case is also studied, revealing an extra term. Our arguments involve the construction of new sub- and super-solutions based on preliminary formal computations on the equation with a moving Dirichlet condition. Finally, convergence to the traveling wave with minimal speed is addressed.

Paper Structure

This paper contains 24 sections, 14 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Enlightening formal computations
  4. Computations involving the minimal wave.
  5. Computations in the moving frame.
  6. The upper estimate
  7. Proof of Proposition \ref{['prop:super-nancy']}
  8. Step 1: a first super-solution.
  9. Step 2: comparison with the solution.
  10. Step 3: conclusion in the delay case $0\leq r<\frac{3}{2}$.
  11. Step 4: conclusion in the advance case $r<0$.
  12. An alternate proof in the advance case ($k>1$)
  13. The lower estimate
  14. A lower bound in the far away zone
  15. Step 1: the first sub-solution.
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $u=u(t,x)$ be the solution of eq with initial data $u_0$ satisfying Assumption ass:initial with $k > -2$. Define Then, there exists a constant $C\geq 0$ such that

Theorems & Definitions (25)

  • Theorem 1.2: Bramson correction when $k> -2$
  • Theorem 1.3: The critical case $k=-2$
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1: Upper estimate
  • Lemma 3.2
  • proof
  • proof : Proof of Proposition \ref{['prop:super-nancy']} when $k>1$
  • ...and 15 more