The Cauchy Problem for Semilinear Parabolic Equations in Besov Spaces

In this paper we first give a unified method by introducing the concept of admissible triplets to study local and global Cauchy problems for semi-linear parabolic equations with a general nonlinear term in different Sobolev spaces. In particular, we establish the local well-posedness and small global well-posedness of the Cauchy problem for semi-linear parabolic equations without the homogeneous condition on the nonlinear term. Our results improve the previously known ones, whilst the proofs are simpler compared with previous ones. Secondly, we establish the local well-posedness and small global well-posedness in Bs p,2 of the Cauchy problem for semi-linear parabolic equations under suitable conditions. Finally, we study the local well-posedness and small global well-posedness in the critical space Bs p,s with n = sp of the Cauchy problem by means of the improved Sobolev inequality established in [22, 23].