聚集-扩散方程与正则或排斥相互作用：高斯与自相似渐近性
Aggregation--Diffusion Equations with Regular or Repulsive Interactions: Gaussian and Self-Similar Asymptotics
- Instytut Matematyczny, Uniwersytet Wrocławski（弗罗茨瓦夫大学数学研究所）
- Mathematical Institute, Tohoku University（东北大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究聚集-扩散方程解的大时间行为，发现正则或温和奇异排斥核导致高斯渐近，而临界对数核导致非线性自相似渐近，并证明自相似解的存在唯一及收敛性。
AI中文摘要：
我们研究了聚集-扩散方程 $$ u_t=\Delta u+\nabla\cdot\bigl(u\\,\nabla K*u\bigr) \qquad \text{在 } \mathbb{R}^d \text{ 中} $$ 的解的大时间行为。我们的主要结果确定了由相互作用核的正则性和奇异性所支配的高斯渐近性与非线性自相似渐近性之间的转变。我们证明，对于正则相互作用核，以及对于足够温和的奇异排斥核，非线性漂移在渐近意义下可忽略不计，解表现出与线性热方程解相同的高斯大时间行为。相反，对于临界对数核 $ K(x)=-\log |x|, $ 相互作用在扩散尺度上持续存在，并导致真正的非线性渐近性。更精确地，对于每个质量 \\(M>0\\)，我们证明了质量为 \\(M\\) 的自相似解的存在性和唯一性，并表明每个质量为 \\(M\\) 且具有有限二阶矩的解在 \\(t\to\infty\\) 时在所有 \\(L^p\\)-范数下收敛到这个自相似解。
英文摘要：
We study the large-time behavior of solutions to the aggregation--diffusion equation $$ u_t=Δu+\nabla\cdot\bigl(u\,\nabla K*u\bigr) \qquad \text{in } \mathbb{R}^d. $$ Our main results identify a transition, governed by the regularity and singularity of the interaction kernel, between Gaussian and nonlinear self-similar asymptotics. We prove that for regular interaction kernels, as well as for sufficiently mild singular repulsive kernels, the nonlinear drift is asymptotically negligible and solutions exhibit the same Gaussian large-time behavior as solutions of the linear heat equation. In contrast, for the critical logarithmic kernel $ K(x)=-\log |x|, $ the interaction persists at the diffusive scale and leads to genuinely nonlinear asymptotics. More precisely, for every mass \(M>0\), we prove the existence and uniqueness of a self-similar solution of mass \(M\) and show that every solution with mass $M$ and with finite second moment converges in all $L^p$-norms to this self-similar solution as \(t\to\infty\).
