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
) ∩ C(
U ) solves the heat equation in UT .
u = max
(ii) Furthermore, if U is connected and there exists a point (x0,t0) ∈ UT
u(x0,t0) = max
u is constant in
Assertion (i) is the maximum principle for the heat equation and (ii)
is the strong maximum principle. Similar assertions are valid with “min”
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