A hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions

Abstract

This paper studies the hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions and logistic source: $u_{t}=-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+\mu u(1-u^k)$, $0=\Delta v+\alpha u^q-\beta v$, $0=\Delta w+\gamma u^r-\delta w$, in a bounded domain $\Omega \subset \mathbb {R}^n$, $n\ge 1$, subject to the non-flux boundary condition. We at first establish the local existence of solutions (the so-called strong $W^{1,p}$-solutions, satisfying the hyperbolic equation weakly and solving the elliptic ones classically) to the model via applying the viscosity vanishing method and then give criteria on global boundedness versus finite- time blowup for them. It is proved that if the attraction is dominated by the logistic source or the repulsion with $\max \{r,k\}>q$, the solutions would be globally bounded; otherwise, the finite-time blowup of solutions may occur whenever $\max \{r,k\}<q$. Under the balance situations with $q=r=k$, $q=r>k$ or $q=k>r$, the boundedness or possible finite-time blowup would depend on the sizes of the coefficients involved.