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and consequently
u
t
.x
;t
/
D
0:
Next we can apply the diffusion equation (
8.16
) to conclude that
u
xx
.x
;t
/
D
u
t
.x
;t
/
D
0;
and therefore
v
xx
.x
;t
/
D
u
xx
.x
;t
/
C
2
D
2 > 0:
This violates property (
8.29
), which must hold at a maximum point of
v
. Hence,
we conclude that
v
must attain its maximum value at the boundary of ˝
T
, i.e.,
v
.x; t /
v
.y; s/
max
.y;s/
2
@˝
T
for all .x; t /
2
˝
T
:
Can
v
reach its maximum value at time t
D
T and for x
2
.0; 1/? No, for the
following reasons this can not be the case. Assume that .x
;T/, with 0<x
<1,
is such that
v
.x; t /
v
.x
;T/
for all .x; t /
2
˝
T
:
Then, according to property (8.32),
v
t
.x
;T/
0;
whichinturnimpliesthat
u
t
.x
;T/
0;
see (
8.34
). Consequently, since
u
satisfies the diffusion equation,
u
xx
.x
;T/
D
u
t
.x
;T/
0;
and it follows that
v
xx
.x
;t
/
D
u
xx
.x
;t
/
C
2
0
C
2 > 0:
This contradicts property (
8.29
) that must be fulfilled at such a maximum point.
This means that
v
.x; t /
v
.y; s/
max
.y;s/
2
T
for all .x; t /
2
˝
T
;
(8.35)
where
T
D f
.0; t /
j
0
t
T
g [ f
.x; 0/
j
0
x
1
g [ f
.1; t /
j
0
t
T
g
:
Let
M
D
max
max
t
0
f.x/
g
1
.t /; max
t
0
g
2
.t /;
max
x
2
.0;1/
