ABSTRACT
This chapter considers the nonhomogeneous Cauchy problem, and assumes some conditions on the coefficients of the operator. It contains the optimal regularity results for the Cauchy problem. The chapter proves some interesting interior Schauder estimates satisfied by the solutions to the differential equation. It also proves that the Cauchy problem admits a unique classical solution. The chapter shows that, the more the data of the Cauchy problem are smoother, the more the solution to such a Cauchy problem is itself smooth. The results concerning existence and regularity of solutions to parabolic problems in the whole space with respect to the Holder norms are well known.
