Global existence and boundedness of classical solutions to a parabolicaparabolic chemotaxis system

This paper deals with the global existence and boundedness of solutions for the chemotaxis system { u t = I u - a a (u I (v) a v) + f (u) , x a ICO , t > 0 , v t = I v - v + u g (u) , x a ICO , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain ICO a R n , with non-negative initial data u 0 a C 0 (ICO A=) and v 0 a W 1 , l (ICO) (for some l > n). The functions I and f are assumed to generalize the chemotactic sensitivity function I (s) = I 0 (1 + beta s) 2 , s aY 0 , with beta aY 0 , I 0 > 0 and the logistic source f (s) = a s - b s 2 , s aY 0 , with a > 0 , b > 0 , respectively. Here g (s) with s aY 0 is a non-negative function. It is proved that the corresponding initialaboundary value problem possesses a unique global classical solution that is uniformly bounded if some technical conditions are fulfilled.