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see, e.g., [28] for further details. Equation (8.6)showsthat
E
1
.t /
D
2
Z
1
0
u
x
.x; t /
dx
for t>0;
which in turns implies that
E
1
.t /
0:
Thus E is a non-increasing function of time t , i.e.,
E
1
.t
2
/
E
1
.t
1
/
for all t
2
t
1
0;
and in particular
Z
1
u
2
.x; t /
dx
Z
1
0
u
2
.x; 0/
dx
D
Z
1
0
f
2
.x/
dx
for t>0:
(8.7)
0
This inequality shows that the integral of the square of the solution of the dif-
fusion equation is bounded by the integral of the square of the initial temperature
distribution f . From a physical point of view, this property seems to be reasonable.
There are no source terms
5
present in our model problem (
8.1
) and the temperature
at x
D
0 and x
D
1 is kept at zero for all time t>0,see(
8.2
). Therefore, it seems
reasonable that the temperature
u
.x; t / at any point x will approach zero as time
increases. Thus, (8.7) is in agreement with our intuition.
6
8.1.2
A Bound on the Derivative
We will now consider one more “energy argument”. Recall that
u
.x; t / can represent
the temperature at position x at time t .Thismeansthat
u
x
represents the “speed”
at which heat flows along the x-axis. From a physical point of view, it seems to be
reasonable that this speed can somehow be bounded by the properties of the initial
condition f ,see(
8.3
). We will now show by mathematical methods that this is
indeed the case. To this end, we define a function E
2
.t / by
5
That is, the function f in (7.1) from Chap. 7 is identical to zero. This function is frequently
referred to as a source term.
6
The alert reader may wonder why this section was titled “Energy Arguments”? There are no
references to energy, since it is used in the theory for heat transfer problems in physics, above. In
fact, there is no strong relation between E
1
.t / and the energy present in the medium for which
(
8.1
)-(
8.3
) is modeling the heat evolution. We will not dwell upon this issue. However, it should
be mentioned that in some models given by PDEs, functions similar to E
1
do represent the energy
present in the underlying physical process. This has led to broader use of the term in mathematics;
it is commonly applied to investigations of functions similar to and of the form of E
1
.
