全抛物型Keller--Segel系统有界性的显式逻辑斯蒂阻尼准则
An Explicit Logistic Damping Criterion for Boundedness in a Fully Parabolic Keller--Segel System
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中文总结 AI 辅助
本文针对有界光滑凸域中的全抛物型Keller--Segel系统，证明了与τ无关的显式系数有界性准则，通过构造新辅助函数完成全局有界性的严格论证。
中文摘要 AI 辅助
我们在有界光滑凸区域中研究全抛物型Keller--Segel系统：\\( u_t=\Delta u-\chi\nabla\\!\u2022(u\nabla v)+\lambda u-\mu u^2 \\)，\\( \tau v_t=\Delta v-v+u \\)。对每个固定的\\( \tau>0 \\)，我们证明\\( \mu>\frac{N\chi}{4} \\)可保证解的全局存在性与时一致有界性。该显式系数充分条件与\\( \tau \\)无关，且不涉及嵌入或最大正则性常数。据我们所知，这是首个在任意空间维度下对所有\\( \tau>0 \\)均保持不变的显式系数有界性准则。证明基于新的辅助比较函数\\( Y_\tau =u+\frac{\chi\tau}{2}|\nabla v|^2-(\tau-1)\Delta v \\)，其对每个\\( \tau>0 \\)满足封闭标量抛物型不等式。当\\( \tau\ge1 \\)时，该不等式给出\\( u \\)的直接点态比较与显式界；当\\( 0<\tau<1 \\)时，它为\\( v \\)提供一致上界。对变换\\( z=e^{-\chi v/2} \\)应用抛物型挤压论证，进而得到\\( v \\)的一致Hölder界；Hölder-Sobolev插值与加权最大\\( L^p \\)正则性随后给出\\( u \\)的充分大\\( p \\)对应的\\( L^p \\)界，标准抛物型平滑性完成论证。
英文摘要
We study the fully parabolic Keller--Segel system \[ u_t=Δu-χ\nabla\!\cdot(u\nabla v)+λu-μu^2, \qquad τv_t=Δv-v+u \] in a bounded smooth convex domain. For every fixed $τ>0$, we prove that \[ μ>\frac{Nχ}{4} \] guarantees global existence and uniform-in-time boundedness. This coefficient-explicit sufficient condition is independent of $τ$ and involves no embedding or maximal-regularity constants. To the best of our knowledge, it is the first coefficient-explicit boundedness criterion that remains unchanged for all $τ>0$ in arbitrary space dimension. The proof is built on a new auxiliary comparison function \[ Y_τ =u+\frac{χτ}{2}|\nabla v|^2-(τ-1)Δv, \] which satisfies a closed scalar parabolic inequality for every $τ>0$. When $τ\ge1$, this inequality yields a direct pointwise comparison and an explicit bound for $u$. When $0<τ<1$, it instead provides a uniform upper bound for $v$. Applying a parabolic squeezing argument to the transform $z=e^{-χv/2}$ then yields a uniform Hölder bound for $v$. Hölder--Sobolev interpolation and weighted maximal $L^p$-regularity subsequently give an $L^p$-bound for $u$ with sufficiently large $p$, and standard parabolic smoothing closes the argument.
