2.3. HEAT EQUATION 57
Infinite propagation speed again. The strong maximum principle im-
plies that if U is connected and u ∈ C1
2(UT
) ∩ C(
¯
U
T
) satisfies
⎧
⎨
⎩
ut − Δu = 0 in UT
u = 0 on ∂U × [0,T ]
u = g on U × {t = 0}
where g ≥ 0, then u is positive everywhere within UT if g is positive some-
where on U. This is another illustration of infinite propagation speed for
disturbances.
An important application of the maximum principle is the following
uniqueness assertion.
THEOREM 5 (Uniqueness on bounded domains). Let g ∈ C(ΓT ), f ∈
C(UT ). Then there exists at most one solution u ∈ C1
2(UT
) ∩ C(
¯T
U ) of the
initial/boundary-value problem
(22)
ut − Δu = f in UT
u = g on ΓT .
Proof. If u and ˜ u are two solutions of (22), apply Theorem 4 to w :=
±(u − ˜). u
We next extend our uniqueness assertion to the Cauchy problem, that
is, the initial-value problem for U =
Rn.
As we are no longer on a bounded
region, we must introduce some control on the behavior of solutions for large
|x|.
THEOREM 6 (Maximum principle for the Cauchy problem). Suppose
u ∈ C1
2(Rn
× (0,T ]) ∩
C(Rn
× [0,T ]) solves
(23)
ut − Δu = 0 in
Rn
× (0,T )
u = g on
Rn
× {t = 0}
and satisfies the growth estimate
(24) u(x, t) ≤
Aea|x|2
(x ∈
Rn,
0 ≤ t ≤ T )
for constants A, a 0. Then
sup
Rn×[0,T ]
u = sup
Rn
g.
