ABSTRACT: In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations with weak data in the homogeneous
spaces. We give a method which can be used to construct local mild solutions of the abstract Cauchy problem inĊ
s,p) by introducing the concept of both admissible quintuplet and compatible space and establishing time-space estimates for
solutions to the linear parabolic type equations. For the small data, we prove that these results can be extended globally
in time. We also study the regularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations
in Ċσ,s,p. As an application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneous Sobolev spaces.
Science in China Series A Mathematics 04/2012; 46(5):641-661. · 0.70 Impact Factor