An Explicit Logistic Damping Criterion for Boundedness in a Fully Parabolic Keller--Segel System

Abstract

We study the fully parabolic Keller--Segel system \[ u_t=\Delta u-\chi\nabla\!\cdot(u\nabla v)+\lambda u-\mu u^2, \qquad \tau v_t=\Delta v-v+u \] in a bounded smooth convex domain. For every fixed $\tau>0$, we prove that \[ \mu>\frac{N\chi}{4} \] guarantees global existence and uniform-in-time boundedness. This coefficient-explicit sufficient condition is independent of $\tau$ 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 $\tau>0$ in arbitrary space dimension. The proof is built on a new auxiliary comparison function \[ Y_\tau =u+\frac{\chi\tau}{2}|\nabla v|^2-(\tau-1)\Delta v, \] which satisfies a closed scalar parabolic inequality for every $\tau>0$. When $\tau\ge1$, this inequality yields a direct pointwise comparison and an explicit bound for $u$. When $0<\tau<1$, it instead provides a uniform upper bound for $v$. Applying a parabolic squeezing argument to the transform $z=e^{-\chi v/2}$ then yields a uniform H\"older bound for $v$. H\"older--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.