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

Abstract

We study the fully parabolic Keller--Segel system \[ u_t=Δu-χ\nabla\!\cdot(u\nabla v)+λu-μu^2, \qquad τv_t=Δv-v+u \] in a bounded smooth convex domain. For every fixed $\tau >0$, we prove that \[ μ>\frac{Nχ}{4} \] 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 \[ Y_τ =u+\frac{χτ}{2}|\nabla v|^2-(τ-1)Δv, \] which satisfies a closed scalar parabolic inequality for every $\tau >0$. When $\tau \ge 1$, 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 Hölder bound for $v$. Hölder--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.