2.3. HEAT EQUATION 59

THEOREM 7 (Uniqueness for Cauchy problem). Let g ∈

C(Rn),

f ∈

C(Rn

×[0, T ]). Then there exists at most one solution u ∈ C1

2(Rn

×(0, T ]) ∩

C(Rn

× [0,T ]) of the initial-value problem

(30)

ut − Δu = f in

Rn

× (0,T )

u = g on

Rn

× {t = 0}

satisfying the growth estimate

(31) |u(x, t)| ≤

Aea|x|2

(x ∈

Rn,

0 ≤ t ≤ T )

for constants A, a 0.

Proof. If u and ˜ u both satisfy (30), (31), we apply Theorem 6 to w :=

±(u − ˜). u

Nonphysical solutions. There are in fact inﬁnitely many solutions of

(32)

ut − Δu = 0 in

Rn

× (0,T )

u = 0 on

Rn

× {t = 0};

see for instance John [J2, Chapter 7]. Each of these solutions besides u ≡ 0

grows very rapidly as |x| → ∞.

There is an interesting point here: although u ≡ 0 is certainly the “physi-

cally correct” solution of (32), this initial-value problem in fact admits other,

“nonphysical”, solutions. Theorem 7 provides a criterion which excludes the

“wrong” solutions. We will encounter somewhat analogous situations in our

study of Hamilton–Jacobi equations and conservation laws, in Chapters 3,

10 and 11.

b. Regularity. We next demonstrate that solutions of the heat equation

are automatically smooth.

THEOREM 8 (Smoothness). Suppose u ∈ C1

2(UT

) solves the heat equa-

tion in UT . Then

u ∈

C∞(UT

).

This regularity assertion is valid even if u attains nonsmooth boundary

values on ΓT .

Proof. 1. Recall from §A.2 that we write

C(x, t; r) = {(y, s) | |x − y| ≤ r, t −

r2

≤ s ≤ t}