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and
D f
.0; t /
j
t
0
g [ f
.x; 0/
j
0
x
1
g [ f
.1; t /
j
t
0
g
:
,(8.35) and the definition (8.33)of
v
imply that
Then, since
T
v
.x; t /
v
.y; s/
M
C
max
.y;s/
2
for all .x; t /
2
˝
T
:
Next, again by (8.33), it follows that
u
.x; t /
v
for all .x; t /
2
˝
T
;
and consequently
u
.x; t /
M
C
for all .x; t /
2
˝
T
and all >0:
This inequality is valid for all >0, and, hence, it follows that
u
.x; t /
M
for all .x; t /
2
˝
T
:
(8.36)
In the argument leading to (8.36), the end “point” T>0of the time interval
Œ0; T , in which we studied the behavior of the solution of the diffusion equation,
was arbitrary, cf. also definitions (
8.25
)-(8.27). Thus, we can choose T as large as
we need! This means that this inequality must hold for any t>0, i.e.,
u
.x; t /
M
for all x
2
Œ0; 1; t > 0:
(8.37)
8.1.8
The Minimum Principle
We have now shown that the solution
u
of our model problem (
8.16
)-(
8.18
)is
bounded by the maximum value M attained by the boundary conditions g
1
and g
2
and the initial temperature distribution f . Can we derive a similar property for the
minimum value achieved by
u
? That is, can we prove that the temperature
u
.x; t /,at
any point .x; t /, is larger than the minimum value achieved by the boundary and ini-
tial conditions? From a physical point of view, this property seems to be reasonable.
Let us now have a look at the mathematics. Consider the function
w
defined by
w
.x; t /
D
u
.x; t /
for all x
2
Œ0; 1; t > 0:
Note that
w
t
D
.
u
/
t
D
u
t
D
u
xx
D
.
u
/
xx
D
w
xx
;