et
be a bounded subset of
.
The
are
functions
.
The
is a positive number. The
is a linear differential
operator


(Operator L 2)

Problem
(Parabolic Dirichlet
problem).
Let us multiply the equations of the problem
(
Parabolic Dirichlet problem
) with
a function
and integrate with respect to
over
.
After integration by parts we
obtain
Definition
(Weak solution of
parabolic Dirichlet problem). The function
is a "weak solution" of the problem
(
Parabolic Dirichlet problem
) if
it
satisfies
To understand the requirement
observe that
with
and compare with the propositions
(
Representation of dual
Sobolev space
) and (
Embedding
of dual Sobolev space
). We see that the
expression
is bounded for
:
Finally
and
hence we need
