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

This paper studies the hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions and logistic source: u t = - χ ∇ · ( u ∇ v ) + ξ ∇ · ( u ∇ w ) + μ u ( 1 - u k ) , 0 = Δ v + α u q - β v , 0 = Δ w + γ u r - δ w , in a bounded domain Ω ⊂ R n , n ≥ 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.