The existence of global and almost-global solutions of heat equations with derivative nonlinear terms is considered for small initial data in the Besov or Triebel--Lizorkin spaces. As an application, the Navier--Stokes equation and the Keller--Segel system of parabolic elliptic type are considered.
"Small solutions for nonlinear heat equations, the Navier-Stokes equation and the Keller-Segel system in Besov and Triebel-Lizorkin spaces." Adv. Differential Equations 18 (7/8) 687 - 736, July/August 2013.