We set
and
and then we also set
and
:
and
add:
Let
then we
have
According to the proposition
(
Energy estimates for the
bilinear form
B
)-1:
thus
We integrate over
and use
to
deduce
We drop the positive term
:
Assume that
.
Let
be the rightmost local maximum of
on
so that
is decreasing on
and
.
Then application of the mean value theorem leads
to
which leads to contradiction with the decreasing behavior of
as we put
.
Therefore
.