Global boundedness in an attraction-repulsion Chemotaxis system with nonlinear productions and logistic source

This paper deals with the attraction-repulsion chemotaxis system with nonlinear productions and logistic source, u(t)=del & sdot;(D(u)del u)-del & sdot; (Phi(u)del v)+del & sdot;(Psi(u)del w)+f(u), v(t)=Delta v+alpha u(k)-beta v, tau w(t)=Delta w+gamma u(l)-delta w, tau is an element of {0,1}, in a bounded domain Omega subset of R-n (n >= 1), subject to the homogeneous Neumann boundary conditions and initial conditions, where D,Phi,Psi is an element of C-2[0,infinity) are nonnegative with D(s)>=(s+1)(p) for s >= 0, Phi(s)<=chi s(q), xi s(g)<=Psi(s)<=zeta s(j), s >= s(0), for s(0)>1, the logistic source satisfies f(s)<= s(a-bs(d)), s>0, f(0)>= 0, and the nonlinear productions for the attraction and repulsion chemicals are described via alpha u(k) and gamma u(l), respectively. When k=l=1, it is known that this system possesses a globally bounded solution in some cases. However, there has been no work in the case k,l > 0. This paper develops the global boundedness of the solution to the system in some cases and extends the global boundedness criteria established by Tian, He, and Zheng (2016) for the attraction-repulsion chemotaxis system.