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.