The Bramson correction in the Fisher-KPP equation: from delay to advance
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 t...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
10.10.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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DOI: | 10.48550/arxiv.2410.07715 |