Global existence and boundedness in a fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities without logistic source

TL;DR

This paper proves the global existence and boundedness of classical solutions for a chemotaxis system with attraction and repulsion, without relying on logistic damping, extending previous results.

Contribution

It demonstrates the existence of global bounded solutions in a chemotaxis model lacking logistic source, which was previously established only with logistic dampening.

Findings

Established global bounded solutions without logistic dampening.

Extended previous results to systems without logistic source.

Provided conditions under which solutions remain bounded.

Abstract

This paper deals with the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, \begin{align*} \begin{cases} u_t=\Delta u-\nabla \cdot (u\chi(v)\nabla v) +\nabla \cdot (u\xi(w)\nabla w), &x \in \Omega,\ t>0,\\[1.05mm] v_t=\Delta v-v+u, &x \in \Omega,\ t>0,\\[1.05mm] w_t=\Delta w-w+u, &x \in \Omega,\ t>0 \end{cases} \end{align*} 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 some conditions. Global existence and boundedness of classical solutions to the system with logistic source have already been obtained by taking advantage of the effect of logistic dampening (J. Math. Anal. Appl.; 2020;489;124153). This paper establishes existence of global bounded classical solutions despite the loss of logistic dampening.