Small solutions for nonlinear heat equations, the Navier-Stokes equation, and the Keller-Segel system in Besov and Triebel-Lizorkin spaces

The existence of globed and almost-globed 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.