(Generic parabolic PDE setup) Let
be a Hilbert space and the Hilbert space
is densely and compactly embedded in
:
where the
refers to the duality with respect to
-topology.
For example,
,
,
compare with the chapter (
Parabolic
PDE
). Let
be a
mapping
measurable with respect to the time parameter
.
For every
the operator
is self adjoint and, uniformly in
,
has the
properties
Let
be
mappings
We are considering the
equation