2.3. HEAT EQUATION 55
Strong maximum principle for the heat equation
2.3.3. Properties of solutions.
a. Strong maximum principle, uniqueness. First we employ the mean-
value property to give a quick proof of the strong maximum principle.
THEOREM 4 (Strong maximum principle for the heat equation). Assume
u ∈ C1
2(UT
) ∩ C(
¯T
U ) solves the heat equation in UT .
(i) Then
max
¯
U
T
u = max
ΓT
u.
(ii) Furthermore, if U is connected and there exists a point (x0,t0) ∈ UT
such that
u(x0,t0) = max
¯
U
T
u,
then
u is constant in
¯
U
t0
.
Assertion (i) is the maximum principle for the heat equation and (ii)
is the strong maximum principle. Similar assertions are valid with “min”
replacing “max”.
Interpretation. So if u attains its maximum (or minimum) at an interior
point, then u is constant at all earlier times. This accords with our strong
intuitive understanding of the variable t as denoting time: the solution will
