64 2. FOUR IMPORTANT LINEAR PDE
THEOREM 11 (Backwards uniqueness). Suppose u, ˜ u
C2(
¯T
U ) solve
(42), (43). If
u(x, T ) = ˜(x, u T ) (x U),
then
u ˜ u within UT .
In other words, if two temperature distributions on U agree at some time
T 0 and have had the same boundary values for times 0 t T , then
these temperatures must have been identically equal within U at all earlier
times. This is not at all obvious.
Proof. 1. Write w := u ˜ u and, as in the proof of Theorem 10, set
e(t) :=
U
w2(x,
t) dx (0 t T ).
As before
(44) ˙(t) e = −2
U
|Dw|2
dx ˙=
d
dt
.
Furthermore
(45)
¨(t) e = −4
U
Dw · Dwt dx
= 4
U
Δwwt dx
= 4
U
(Δw)2
dx by (41).
Now since w = 0 on ∂U,
U
|Dw|2
dx =
U
wΔw dx

U
w2
dx
1/2
U
(Δw)2
dx
1/2
.
Thus (44) and (45) imply
(˙(t))2
e = 4
U
|Dw|2
dx
2

U
w2
dx 4
U
(Δw)2
dx
= e(t)¨(t). e
Previous Page Next Page