We consider a second order linear parabolic nonhomogeneous i.b.v. problem in [0,T]\(\times {\bar \Omega}\) where \(\Omega\) is a bounded regular open set in \({\mathbb{R}}^ n\). We establish an optimal regularity theorem, proving that the derivatives \(u_ t\), \(D_{ij}u\) of the solution u are continuous in [0,T]\(\times {\bar \Omega}\) and \(\alpha\)-Hölder continuous with respect to the space variables, provided the data satisfy the necessary smoothness and regularity assumptions.
The linear results are then used to study local existence, regularity and dependence on the initial value of the solutions of general fully nonlinear second order parabolic problems. In particular, we show that a large class of fully nonlinear equations generates local semiflows on suitable subsets of \(C^{2,\alpha}({\bar \Omega})\).