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Andrew Zhao

Homework 5

1

Exercise 4. Let D ⊂ Rd be a bounded domain with a smooth boundary and consider the
heat equation with Neumann boundary conditions:

ut − ∆u = f (x)
in D
(1a)
∇u · n
ˆ=0
on ∂D
(1b)
From the divergence theorem and (1b) we find that
Z
Z
∇ · (∇u) dx =
(∇u · n
ˆ ) dS(x)
D
Z∂D
=
0 · dS(x)
∂D

= 0.
We can rewrite (1a) as ∆u = ut − f (x) and, noting that the Laplacian ∆ ≡ ∇ · ∇, take
volume integrals over D on all terms such that
Z
Z
Z
∇ · (∇u) dx =
ut dx −
f (x) dx
D
D
D
Z
Z
f (x) dx
ut dx −
0=
D
Z
ZD
f (x) dx =
ut dx .
D

D

If f (x) is strictly positive (i.e. always a source, never a sink), then we find with the boundary
conditions (1b) that
Z
Z
f (x) dx > 0 ⇒

ut dx > 0 .
D

D

Thus the solution to (1), u = u(t, x), is strictly growing over time in D. From this we can
claim that
Z
u(t, x) dx = +∞ .
lim
t→∞

D