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w
x
.x; t /
¤
0
or
w
t
.x; t /
¤
0
for all .x; t /
2
˝
T
;
(8.30)
or
w
xx
.x; t / > 0
or
w
tt
.x; t / > 0
for all .x; t /
2
˝
T
;
(8.31)
then the maximum value of
w
must be attained at the boundary @˝
T
of ˝
T
.This
observation is, as we will see below, the main ingredient of the analysis presented
in this section. Note also that, analogously to (8.24), if the maximum is achieved for
t
D
T ,say,at.x
;T/,then
w
must satisfy the inequality
13
w
t
.x
;T/
0:
(8.32)
As mentioned above, we will not go through any derivations of these properties
for the maxima of smooth functions defined on closed sets. Further details can be
found in, e.g., Apostol [4] or Edwards and Penney [13]. Instead we will now turn
our attention toward the heat equation and, more specifically, to how these results
can be utilized to increase our insight into this problem.
Assume that
u
is a smooth solution of (
8.16
)-(
8.1
8
).
For an arbitrary time T>0
we want to study the behavior of
u
on the closed set ˝
T
defined in (8.27). Assume
that
u
achieves its maximum value at an interior point .x
;t
/
2
˝
T
.From(
8.16
),
it follows that
u
t
.x
;t
/
D
u
xx
.x
;t
/;
andby(
8.28
) we conclude that
u
xx
.x
;t
/
D
0:
Thus we can not use (
8.29
) directly to exclude the possibility that
u
attains its
maximum value at an interior point. We have to take a detour!
To this end we define a family of auxiliary functions
f
v
g
>0
by
v
.x; t /
D
u
.x; t /
C
x
2
for >0;
(8.33)
where
u
solves (
8.16
)-(
8.18
). Note that
v
t
.x; t /
D
u
t
.x; t /;
(8.34)
v
xx
.x; t /
D
u
xx
.x; t /
C
2 >
u
xx
.x; t /:
Now, if
v
achieves its maximum at an interior point, say .x
;t
/
2
˝
T
,then
property (
8.28
) implies that
v
t
.x
;t
/
D
0;
13
Similar properties must hold at any maximum point located at the boundary @˝
T
of ˝
T
, i.e., at
maximum points with coordinates of the form .x
;0/, .0; t
/ or .1; t
/. However, in the present
analysis we will only need the inequality (8.32).