Global boundedness in the higher-dimensional fully parabolic chemotaxis with weak singular sensitivity and logistic source
Minh Le
TL;DR
The paper analyzes a higher-dimensional fully parabolic chemotaxis system with weak singular sensitivity $k \in (0,1)$ and logistic damping in a smooth bounded domain $\Omega \subset \mathbb{R}^n$ with $n \ge 3$ under homogeneous Neumann boundary conditions. It proves that there exists a threshold $\mu_0=\mu_0(n,k,r,\chi,\beta)$ such that for all $\mu>\mu_0$, the system has a unique nonnegative classical solution $(u,v)$ that remains globally bounded, with $u$ and $v$ strictly positive. The core technique is an energy method based on the functional $y(t)=\int_{\Omega} u^p + \int_{\Omega} u^p v^{-q} + \int_{\Omega} v^{p+1}$ for carefully chosen $p>0$ and $q>0$, together with two key estimates (Lemmas \ref{l1} and \ref{LK-1}) to derive $L^p$-bounds and higher regularity. A critical challenge is that the logistic term breaks mass conservation, which can drive $v$ toward zero and blow up the cross-diffusion term; the authors overcome this with the energy framework, parabolic regularity, and semigroup estimates to achieve global boundedness in arbitrary dimensions, extending prior two-dimensional results.
Abstract
We consider the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain $Ω\subset \mathbb{R}^n$ with $n \geq 3$: \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - μu^2, & \text{in } Ω\times (0,T_{\rm max}), v_t = Δv - αv + βu, & \text{in } Ω\times (0,T_{\rm max}), \end{cases} \end{equation*} where $k \in (0,1)$, and $χ, r, μ, α, β$ are positive parameters. In this paper, we demonstrate that for suitably smooth initial data, the problem admits a unique nonnegative classical solution that remains globally bounded in time when $μ$ is sufficiently large.