Existence of weak solution for parabolic Dirichlet problem.

roposition

(Existence of weak solution for the parabolic Dirichlet problem).
There exists a weak solution of the problem
(
Parabolic Dirichlet problem
).

Proof

The sequence
of approximate Galerkin solutions given by the equations
(
Galerkin problem
) satisfies the
relationships
for any
from the linear span of
,
.
According to the estimates of the proposition
(
Energy
estimates for the Galerkin approximate solution
) and the proposition
(
Weak compactness of bounded
set
) there is a subsequence of
that converges weakly in
to some
and
converges weakly in
.
Such convergence is what we need to pass to the limit in (*) and
obtain
for any
.

Proposition

(Uniqueness of weak solution for the parabolic Dirichlet
problem). The weak solution of the problem
(
Parabolic Dirichlet problem
) is
unique.