发表机构
Faculty of Engineering, University of Fukui; National Institute of Technology(KOSEN), Fukui College（福井大学工学部; 福井国立高等专门学校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究吸引-排斥趋化系统稳定稳态小扰动的大时间行为，建立衰减估计并证明非线性扰动渐近由线性化演化逼近，其前导项为有效热流，在一维可积情形下由初始扰动总质量决定高斯轮廓。
AI 中文摘要
我们研究了在$n$维欧几里得空间中，吸引-排斥趋化系统稳定常数稳态的小扰动的大时间行为。我们考虑一维空间中的可积扰动，以及在更高维度中属于具有适当指数的Lebesgue空间的扰动。我们首先在整个允许的Lebesgue指数范围内建立衰减估计，并证明非线性扰动渐近地由相应的线性化演化逼近。然后，我们将线性化演化的前导项识别为有效热流，其扩散系数由系统参数明确确定。因此，当可积指数严格小于空间维度时，前导渐近轮廓由热方程控制。在临界端点处，非线性扰动与有效热流之间的差异在自然抛物型标度下消失。在一维可积情形中，这产生了由初始扰动的总质量确定的高斯轮廓。
英文摘要
We investigate the large-time behavior of small perturbations of stable constant steady states for an attraction-repulsion chemotaxis system in $n$-dimensional Euclidean space. We consider integrable perturbations in one space dimension and, in higher dimensions, perturbations belonging to Lebesgue spaces with suitable exponents. We first establish decay estimates throughout the full admissible range of Lebesgue exponents and show that the nonlinear perturbation is asymptotically approximated by the corresponding linearized evolution. We then identify the leading term of the linearized evolution as an effective heat flow whose diffusion coefficient is explicitly determined by the parameters of the system. Consequently, when the integrability exponent is strictly smaller than the space dimension, the leading asymptotic profile is governed by the heat equation. At the critical endpoint, the difference between the nonlinear perturbation and the effective heat flow vanishes under the natural parabolic scaling. In the one-dimensional integrable case, this yields a Gaussian profile determined by the total mass of the initial perturbation.
