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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

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