We study traveling waves of the KPP equation in the half-space with Dirichlet boundary conditions. We show that minimal-speed waves are unique up to translation and rotation but faster waves are not. We represent our waves as Laplace transforms of martingales associated to branching Brownian motion in the half-plane with killing on the boundary. We thereby identify the waves’ asymptotic behavior and uncover a novel feature of the minimal-speed wave <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>. Far from the boundary, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> converges to a logarithmic shift of the 1D wave w of the same speed: <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\displaystyle \lim _{y \rightarrow \infty } \Phi \big (x + \tfrac{1}{\sqrt{2}}\log y, y\big ) = w(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <munder> <mo movablelimits="true">lim</mo> <mrow> <mi>y</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <mi mathvariant="normal">Φ</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>x</mi> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> <mo>log</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mstyle> </math></EquationSource> </InlineEquation>.

KPP Traveling Waves in the Half-Space

期刊论文
发表日期：

2025年10月3日

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Julien Berestycki,
Cole Graham,
Yujin H. Kim,
Bastien Mallein

摘要

We study traveling waves of the KPP equation in the half-space with Dirichlet boundary conditions. We show that minimal-speed waves are unique up to translation and rotation but faster waves are not. We represent our waves as Laplace transforms of martingales associated to branching Brownian motion in the half-plane with killing on the boundary. We thereby identify the waves’ asymptotic behavior and uncover a novel feature of the minimal-speed wave $\Phi $ $Φ$ . Far from the boundary, $\Phi $ $Φ$ converges to a logarithmic shift of the 1D wave w of the same speed: $\displaystyle \lim _{y \rightarrow \infty } \Phi \big (x + \tfrac{1}{\sqrt{2}}\log y, y\big ) = w(x)$ $lim_{y \to \infty} Φ (x + \frac{1}{\sqrt{2}} log y, y) = w (x)$ .