Global existence and boundedness of a chemotaxis model with indirect production and general kinetic function

This paper is devoted to the chemotaxis model with indirect production and general kinetic functionut {u(t) = Delta u - chi del center dot (u del v) + f(u), x is an element of Omega, t > 0, vt = Delta v - v+w, x is an element of Omega, t > 0, tau w(t) + lambda w = g(u), x is an element of Omega, t > 0, in a bounded domain Omega subset of R-n(n <= 3)with smooth boundary partial differential partial derivative Omega, where chi, tau, lambda are given positive parameters,fandgare known functions. We find several explicit conditions involving the kinetic functionf,g, the parameters chi, lambda, and the initial data vertical bar vertical bar u(0)vertical bar vertical bar L-(Omega)(1) to ensure the global-in-time existence and uniform boundedness for the corresponding 2D/3D Neumann initial-boundary value problem. Particularly, when f equivalent to 0, andgis a linear function, the global bounded classical solutions to the corresponding 2D Neumann initial-boundary value problem with arbitrarily large initial data and chemotactic sensitivity are established. Our results partially extend the results of Hu and Tao (Math Models Methods Appl Sci 26:2111-2128, 2016), Tao and Winkler (J Eur Math Soc 19:3641-3678, 2017), etc.