Persistence of mass in a chemotaxis system with logistic source

Abstract

This paper studies the dynamical properties of the chemotaxis systemx ∈ , t > 0, ( ) under homogeneous Neumann boundary conditions in bounded convex domains ⊂ R n , n ≥ 1, with positive constants χ , r and μ. Numerical simulations but also some rigorous evidence have shown that depending on the relative size of r, μ and | |, in comparison to the well-understood case when χ = 0, this problem may exhibit quite a complex solution behavior, including unexpected effects such as asymptotic decay of the quantity u within large subdomains of .The present work indicates that any such extinction phenomenon, if occurring at all, necessarily must be of spatially local nature, whereas the population as a whole always persists. More precisely, it is shown that for any nonnegative global classical solution (u, v) of ( ) with u ≡ 0 one can find m > 0 such thatThe proof is based on an, in this context, apparently novel analysis of the functional ln u, deriving a lower bound for this quantity along a suitable sequence of times by appropriately exploiting a differential inequality for a suitable linear combination of ln u, u and v 2 .

A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source

“…Going beyond the basic knowledge of above boundedness results, some important findings were given by many authors which assert that the interaction effects between cross-diffusion and cell kinetics may result in quite a colorful dynamics (see e.g. Winkler et al [37,54,53], Galakhov et al [19], Zheng [61]). For example, Osaki etal.…”

Parabolic–elliptic chemotaxis model with space–time dependent logistic sources on RN. II. Existence, uniqueness, and stability of strictly positive entire solutions

“…Recently, several studies have been concerned with establishing adequate conditions on the chemotaxis sensitivity χ and other parameters in (1.2) to ensure the existence of time global solutions and the stability of equilibria solutions. In this regard, we refer to [12,13,22,27,30,31]. The feature of solutions of (1.2) in the presence of logistic type sources still remains a very interesting problem.…”

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

“…In contrast to the rich knowledge on boundedness, convergence and other dynamical properties for (1.1) and its variants, understanding the qualitative or quantitative properties even of bounded solutions to chemotaxis problems seems much less developed. In this direction, a work was considered by Tao and Winkler in [28] to show the mass persistence phenomenon for (1.1), i.e, for any supposedly given global classical and bounded nontrivial solution (u, v) of (1.1), there is m * > 0 such that u(t) L 1 ≥ m * for all t > 0. To our best knowledge, there seems no work on how boundedness or upper bounds of solutions of (1.1) depends on the system parameters, say, χ, µ or r. In this paper, we aim as a first step to study chemotaxis effect vs logistic damping on boundedness for the minimal chemotaxis-logistic model (1.1) in 2-D. We do so partially because all solutions in 2-D are global and bounded by [23,38].…”