44 2. FOUR IMPORTANT LINEAR PDE
2.3. HEAT EQUATION
Next we study the heat equation
(1) ut − Δu = 0
and the nonhomogeneous heat equation
(2) ut − Δu = f,
subject to appropriate initial and boundary conditions. Here t 0 and
x ∈ U, where U ⊂
is open. The unknown is u :
U × [0, ∞) → R, u =
u(x, t), and the Laplacian Δ is taken with respect to the spatial variables x =
(x1,...,xn): Δu = Δxu =
uxixi . In (2) the function f : U×[0, ∞) → R
A guiding principle is that any assertion about harmonic functions yields
an analogous (but more complicated) statement about solutions of the heat
equation. Accordingly our development will largely parallel the correspond-
ing theory for Laplace’s equation.
Physical interpretation. The heat equation, also known as the diﬀusion
equation, describes in typical applications the evolution in time of the density
u of some quantity such as heat, chemical concentration, etc. If V ⊂ U is
any smooth subregion, the rate of change of the total quantity within V
equals the negative of the net flux through ∂V :
u dx = −
F being the flux density. Thus
(3) ut = − div F,
as V was arbitrary. In many situations F is proportional to the gradient
of u but points in the opposite direction (since the flow is from regions of
higher to lower concentration):
F = −aDu (a 0).
Substituting into (3), we obtain the PDE
ut = a div(Du) = aΔu,
which for a = 1 is the heat equation.
The heat equation appears as well in the study of Brownian motion.