Absence of blow-up in a fully parabolic chemotaxis system with weak singular sensitivity and logistic damping in dimension two
Minh Le
TL;DR
The study investigates blow-up behavior for a two-dimensional fully parabolic chemotaxis system with weak singular sensitivity $\frac{1}{v^k}$ ($k\in(0,1)$) and logistic damping. It develops a sequence of a priori estimates, including an $L^1$ bound, an $L\log L$ bound for $u$, $L^p$ bounds, and higher-order control of $\nabla v$ via parabolic regularity, facilitated by energy functionals $y(t)$ and $z(t)$. These ingredients enable a Gronwall-based argument that yields global existence and uniform boundedness of solutions, with $u$ uniformly bounded and both $u,v$ remaining strictly positive. The results extend known blow-up absence from the parabolic-elliptic case to the fully parabolic model in 2D, highlighting the robust blow-up-preventing role of logistic damping in the presence of weak sensitivity.
Abstract
It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain \(Ω\subset \mathbb{R}^2\): \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - μu^2, \qquad &\text{in } Ω\times (0,T_{\rm max}), v_t = Δv - αv + βu, \qquad &\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. Known results have already established the same conclusion for the parabolic-elliptic case. Here, we complement these findings by extending the result to the fully parabolic case.