化学趋化系统中临界通量限制下的质量阈值问题

本文研究了由 Kohatsu 和 Senba (2025) 提出的通量受限化学趋化系统\n\begin{equation*}\begin{cases}\n u_t = \Delta u -\nabla\cdot(u|\nabla v|^{\alpha-2}\nabla v),\n\ \:\:,0=\Delta v + u,\n\end{cases}\end{equation*}\n posed in the unit ball of $\mathbb{R}^N$ for some $N\geq2$, subject to no-flux and homogeneous Dirichlet boundary conditions. Due to precedents, e.g., Tello (2022) and Winkler (2022), the exponent $\alpha = \frac{N}{N-1}$ is the threshold for finite-time blow-up under symmetry assumptions. We further find that under the framework of radially symmetric solutions, the system with critical flux limitation admits a globally bounded weak solution if and only if initial mass is strictly less than $\omega_N \big(\frac{N^2}{N-1}\big)^{N-1}$, where $\omega_N$ denotes the measure of the unit sphere $\mathbb{S}^{N-1}$. Asymptotic behaviors are also considered.\n\n由于 Tello (2022) 和 Winkler (2022) 等已有研究，指数 $\alpha = \frac{N}{N-1}$ 在对称性假设下为有限时间 blow-up 的临界值。我们进一步发现，在径向对称解的框架下，具有临界通量限制的系统若且仅若初始质量严格小于 $\omega_N \big(\frac{N^2}{N-1}\big)^{N-1}$（其中 $\omega_N$ 为单位球 $\mathbb{S}^{N-1}$ 的面积），则 admits a globally bounded weak solution。也讨论了渐近行为。