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 ⊂

Rn

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 =

∑n

i=1

uxixi . In (2) the function f : U×[0, ∞) → R

is given.

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 :

d

dt

V

u dx = −

∂V

F ·

ν

dS,

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.