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).