2.3. HEAT EQUATION 63
THEOREM 10 (Uniqueness). There exists only one solution u ∈ C1
2(
¯T
U )
of the initial/boundary-value problem (40).
Proof. 1. If ˜ u is another solution, w := u − ˜ u solves
(41)
wt − Δw = 0 in UT
w = 0 on ΓT .
2. Set
e(t) :=
U
w2(x,
t) dx (0 ≤ t ≤ T ).
Then
˙(t) e = 2
U
wwt dx ˙=
d
dt
= 2
U
wΔw dx
= −2
U
|Dw|2
dx ≤ 0,
and so
e(t) ≤ e(0) = 0 (0 ≤ t ≤ T ).
Consequently w = u − ˜ u ≡ 0 in UT .
Observe that the foregoing is a time-dependent variant of the proof of
Theorem 16 in §2.2.5.
b. Backwards uniqueness. A rather more subtle question asks about
uniqueness backwards in time for the heat equation. For this, suppose u
and ˜ u are both smooth solutions of the heat equation in UT , with the same
boundary conditions on ∂U:
(42)
ut − Δu = 0 in UT
u = g on ∂U × [0,T ],
(43)
˜t u − Δ˜ u = 0 in UT
˜ u = g on ∂U × [0,T ],
for some function g. Note carefully that we are not supposing u = ˜ u at time
t = 0.
