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