Large time behavior of solution to a fully parabolic chemotaxis system with singular sensitivity and logistic source

This paper presents the large time behavior of solution to the fully parabolic chemotaxis system with singular sensitivity and logistic source {u(t)=del.(D(u)del u)-chi del.(u/v kappa del v)+mu u-mu u(2), x is an element of Omega,t > 0, v(t)=Delta v-v+u, x is an element of Omega,t > 0, with homogeneous Neumann boundary condition in a convex smooth bounded domain Omega subset of R-n, n >= 2, where chi > 0, mu > 0 and kappa is an element of(0, 1/2)boolean OR(1/2, 1), D(u) is supposed to satisfy the following property D(u)>=(u+1)(alpha) with alpha > 0. One can find a positive constant m(*) such that integral(Omega;)u >= m(*) for all t >= 0. Apart from that, it is shown that the solution is globally bounded. Furthermore, it is asserted that the solution exponentially converges to the steady state (1,1) as t ->infinity. (c) 2022 Elsevier Ltd. All rights reserved.