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