Parabolic comparison revisited and applications

Abstract

We consider the Cauchy-Dirichlet problem ${\partial }_{t}u-F(t,x,u,Du,D^{2}u)=0on(0,T)\times {\mathbb{R}}^n$ in viscosity sense. Comparison is established for bounded semi-continuous (sub-/super-)solutions under structural assumption (3.14) of the User's Guide plus a mild condition on $F$ such as to cope with the unbounded domain. Comparison on $(0,T]$, space-time regularity and existence are also discussed. Our analysis passes through an extension of the parabolic theorem of sums which appears to be useful in its own right.