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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).
 
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