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.