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.