GLOBAL EXISTENCE AND DECAY RATES OF THE SOLUTIONS FOR A CHEMOTAXIS SYSTEM WITH LOTKA-VOLTERRA TYPE MODEL FOR CHEMOATTRACTANT AND REPELLENT

We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish L-q time-decay for the linear homogeneous system by using a Fourier transform and finding Green's matrix. Then, we find L-q time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimates.