We start with

The last a priori estimate shows that the parabolic solution becomes infinitely
smoother than its initial data (- we
"gain" infinitely many *s*-derivatives), and at
the same time these higher derivatives decay faster as .

Alternatively, we can work with the energy method.
Multiply (para.1) by *u* and integrate to get

and in general

successive integration of (para.7) yields (para.5).

Thu Jan 22 19:07:34 PST 1998