Bramson 修正与 $\mathbb Z^d$ 上 Fisher-KPP 方程临界波的收敛性
Bramson correction and convergence to the critical wave for Fisher-KPP equations on $\mathbb Z^d$
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中文总结 AI 辅助
本文研究 $\mathbb Z^d$ 上 Fisher-KPP 方程，证明波前对数延迟、过渡区宽度有界及解收敛到临界行波，方法为加权估计与乘积下解。
中文摘要 AI 辅助
我们考虑 $\mathbb Z^d$（$d\geq2$）上具有最近邻扩散和非零有限支撑初始数据的 Fisher-KPP 方程。我们证明了波前沿每个有符号坐标轴的传播存在对数延迟，并表明过渡区域具有一致有界的宽度。在围绕轴的每个固定宽度的半管中，解收敛到最小速度格点行波的平移，且相位有界。我们还获得了每个方向上带对数修正的上界。证明使用了加权估计、倾斜随机游走的界以及乘积下解。不假设反应项的凹性。
英文摘要
We consider Fisher-KPP equations with nearest-neighbor diffusion on $\mathbb Z^d$, $d\geq2$, with nonzero finitely supported initial data. We prove a logarithmic delay of the front along each signed coordinate axis and show that the transition region has uniformly bounded width. On every fixed-width half-tube around an axis, the solution converges to translates of the minimal-speed lattice traveling wave, with a bounded phase. We also obtain an upper bound with a logarithmic correction in every direction. The proof uses weighted estimates, bounds for tilted random walks, and a product lower solution. Concavity of the reaction is not assumed.
发表机构
- School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）
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