<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3231_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(-r \ln t\)</EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3231_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(r < \frac{3}{2}\)</EquationSource> </InlineEquation>, which corresponds to a logarithmic delay when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3231_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(0< r < \frac{3}{2}\)</EquationSource> </InlineEquation> and a logarithmic advance when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3231_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(r < 0\)</EquationSource> </InlineEquation>. The critical case <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3231_Article_IEq5.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(r=\frac{3}{2}\)</EquationSource> </InlineEquation> is also studied, revealing an extra <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3231_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {O}}(\ln \ln t)\)</EquationSource> </InlineEquation> 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.</p>

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

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2025年7月10日

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Matthieu Alfaro,
Thomas Giletti,
Dongyuan Xiao

摘要

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 \(r=\frac{3}{2}\) 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.