Akademik Veri Yönetim Sistemi
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, cilt.69, 2023 (SCI-Expanded, Scopus)
This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, {u(t) = Delta u - chi del center dot (u/v del v) + u(a(t, x) - b(t, x)u), x is an element of I-2 0 = Delta v - mu v + nu u, x is an element of I-2 partial derivative u/partial derivative n = partial derivative v/partial derivative v = 0, x is an element of Omega I-2, x is an element of partial derivative n (0.1) where Omega subset of R-N is a smooth bounded domain, a(t, x) and b(t, x) are positive smooth functions, and chi, mu and nu are positive constants. In recent years, it has been drawn a lot of attention to the question of whether logistic kinetics prevents finite-time blow-up in various chemotaxis models. In the very recent paper (Kurt and Shen, 2021), we proved that for every given nonnegative initial function 0 (sic) u(0) is an element of C-0((Omega) over bar over bar ) and s is an element of R, (0.1) has a unique globally defined classical solution (u(t, x, s, u(0)), v(t, x, s, u(0))) with u(s, x, s, u(0)) = u(0)(x), which shows that, in any space dimensional setting, logistic kinetics prevents the occurrence of finite-time blow-up even for arbitrarily large x. In Kurt and Shen (2021), we also proved that globally defined positive solutions of (0.1) are uniformly bounded under the assumption { mu chi 2 if 0< x <= 2 ainf > 4 mu(x -1) ifx > 2. (0.2) In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption (0.2). Among others, we provide some concrete estimates for integral ohm u-p and integral ohm uq for some p > 0 and q > 2N and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a "rectangular" type bounded invariant set (in Lq) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution (u*(t, x), v*(t, x)), which is periodic in t if a(t, x) and b(t, x) are periodic in t and is independent of t if a(t, x) and b(t, x) are independent of t. (C) 2022 Elsevier Ltd. All rights reserved.