arXiv subjects

Jianlu Yan

Publications and source records attributed to Jianlu Yan.

3 recordsLinked to original sources

arXiv2025

Mass threshold for global existence in chemotaxis systems with critical flux limitation

This paper investigates the flux-limited chemotaxis system, proposed by Kohatsu and Senba~(2025), \begin{equation*} \begin{cases} u_t = \Delta u -\nabla\cdot(u|\nabla v|^{\alpha-2}\nabla v),\\ \:\:0=\Delta v + u, \end{cases} \end{equation*} 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.

math.AP↗

arXiv2020

When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions?

We construct global generalized solutions to the chemotaxis system \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + λ(x) u - μ(x) u^κ,\\ v_t = Δv - v + u \end{cases} \end{align*} in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \geq 2$, for certain choices of $λ, μ$ and $κ$. Here, inter alia, the selections $μ(x) = |x|^α$ with $α< 2$ and $κ= 2$as well as $μ\equiv μ_1 > 0$ and $κ> \min\{\frac{2n-2}{n}, \frac{2n+4}{n+4}\}$ are admissible (in both cases for any sufficiently smooth $λ$). While the former case appears to be novel in general, in the two- and three-dimensional setting, the latter improves on a recent result by Winkler (Adv. Nonlinear Anal. 9 (2019), no. 1, 526-566), where the condition $κ> \frac{2n+4}{n+4}$ has been imposed. In particular, for $n = 2$, our result shows that taking any $κ> 1$ suffices to exclude the possibility of collapse into a persistent Dirac distribution.

math.AP↗

arXiv2018

Global generalized solutions to a nonlinear Keller-Segel equation with singular sensitivity

We consider the chemotaxis system \begin{eqnarray*} \begin{cases} \begin{array}{lll} \medskip u_t =Δu^m - \nabla(\frac{u}{v}\nabla v),&{} x\inΩ,\ t>0, \medskip v_t =Δv -uv,&{}x\inΩ,\ t>0, \medskip \frac{\partial u}{\partial ν}=\frac{\partial v}{\partialν}=0,&{}x\in\partialΩ,\ t>0, \medskip u(x,0)=u_0(x),\ v(x,0)=v_0(x), &{}x\inΩ, \end{array} \end{cases} \end{eqnarray*} in a smooth bounded domain $Ω\subset \mathbb{R}^n$, $n\geq2$. In this work it is shown that for all reasonably regular initial data $u_0\geq0$ and $v_0>0$, the corresponding Neumann initial-boundary value problem possesses a global generalized solution provided that $m>1+\frac{n-2}{2n}$.

math.AP↗