ABSTRACT
This chapter focuses on the analysis of parabolic equations in the whole space and in bounded domains. It deals with the nonhomogeneous Cauchy problem. The chapter explains the main properties of the heat kernel and uses such results to prove that the function is the unique classical solution to the Cauchy problem. It proves two equivalent characterizations of the Holder spaces, which are given in terms of the Gauss-Weierstrass semigroup. The chapter explains the fundamental optimal Schauder estimates for solutions to the Cauchy problem. It also proves the existence and uniqueness of a classical solution to the Cauchy problem. The Gauss-Weierstrass semigroup is not strongly continuous.
