Lecture 10, Oct 26, 13:10-14:00 regularity of the solution of heat equation: C^\infy in x,t when U bounded. real analytic in x, not real analytic in t (strong max principle: cannot predict future events from past, distubence from boundary at future time contributes to later events). When U is the whole space, real analytic in x,t (the relevant solution only) real analytic in t since there is no boundary data imposed, hence no "disturbance from boundary". (Ref: John) Left many details, lack of time.