Global existence and uniform boundedness to a bi-attraction chemotaxis system with nonlinear indirect signal mechanisms

In this paper, we study the following quasilinear chemotaxis system ut = increment u - chi V center dot (phi(u)Vv) - xi V center dot (psi(u)Vw) + f(u) , x E S2 , t > 0 , 0 = increment v - v + v gamma 1 1 , 0 = increment v1 - v1 + u gamma 2 , xE S2 , t>0 , 0= increment w - w + w gamma 3 1 , 0 = increment w1 - w1 + u gamma 4 , xE S2 , t>0 , in a smoothly bounded domain S2 c Rn(n > 1) with homogeneous Neumann boundary conditions, where phi(rho) < rho(rho +1)theta-1 , psi(rho) < rho(rho +1)l-1 and f(rho) < a rho - b rho s for all rho > 0 , and the parameters satisfy a , b , chi, xi, gamma 2 , gamma 4 > 0 , s > 1 , gamma 1 , gamma 3 > 1 and theta, l E R. It has been proven that ifs > max{gamma 1 gamma 2 + theta, gamma 3 gamma 4 + l) , then the system has a nonnegative classical solution that is globally bounded. The boundedness condition obtained in this paper relies only on the power exponents of the system, which is independent of the coefficients of the system and space dimension n. In this work, we generalize the results established by previous researchers.