Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 48, pp. 1-23.

Title: Cauchy problems for chemotaxis systems with chemo attractant and repellent
       
Authors: Aesha Lagha (Zawia University, Zawia, Libya})
         Harumi Hattori (West Virginia Univ., Morgantown, WV, USA)

Abstract: 
 We study the existence of global solutions and their asymptotic behavior
for a chemotaxis system with chemo-attractant and repellent in three
dimensions. To accomplish this, we use the Fourier transform and energy
method. First, we establish L<sup>q</sup> time-decay for the linear homogeneous
system using Fourier transform and finding Green's matrix. Then, we
find  L<sup>q</sup> time-decay for the nonlinear system using the Duhamel's
principle and the time-weighted estimates.

Submitted September 4, 2021. Published July 05, 2022.
Math Subject Classifications: 35Q31, 35Q35, 35Q92, 76N10.
Key Words: Energy method; a priori estimates; Fourier transform; time-decay rates.