Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 71, pp. 1-10.

Title: Existence of bounded global solutions for fully parabolic attraction-repulsion 
       chemotaxis systems with signal-dependent sensitivities and without logistic source
       
Authors: Yutaro Chiyo (Tokyo Univ. of Science, Japan)
         Masaaki Mizukami (Kyoto Univ. of Education, Japan)
         Tomomi Yokota (Tokyo Univ. of Science, Japan)

Abstract:
This article concerns the parabolic attraction-repulsion
chemotaxis system with signal-dependent sensitivities
$$\displaylines{
u_t=\Delta u-\nabla \cdot (u\chi(v)\nabla v) +\nabla \cdot (u\xi(w)\nabla w),
 \quad x \in \Omega,\; t>0,  \cr
 v_t=\Delta v-v+u, \quad x \in \Omega,\; t>0, \cr
 w_t=\Delta w-w+u, \quad x \in \Omega,\; t>0
 }$$
under homogeneous Neumann boundary conditions and initial conditions,
where $\Omega \subset \mathbb{R}^n$ $(n \ge 2)$ is a bounded domain with
smooth boundary, $\chi, \xi$ are functions satisfying certain conditions.
Existence of bounded global classical solutions to the system with logistic source
and logistic damping have  been obtained in [1].
This article establishes the existence of global bounded classical solutions
with logistic damping.

Submitted April 8, 2021. Published September 10, 2021.
Math Subject Classifications: 35A01, 35Q92, 92C17.
Key Words: Chemotaxis; attraction-repulsion; existence; boundedness.