Math 425, Spring 2011
Jerry L. Kazdan
Problem Set 9
D
UE
: Thursday April 7 [
Late papers will be accepted until 1:00 PM Friday
].
1.
a)
In a bounded region
Ω
⊂
R
n
, let
u
(
x
,
t
)
satisfy the modified heat equation
u
t
=
Δ
u
+
cu
,
where
c
is a constant
,
(1)
as well as the initial and boundary conditions
u
(
x
,
0
) =
f
(
x
)
,
in
Ω
with
u
(
x
,
t
) =
0 for
x
∈
∂Ω
,
t
≥
0
.
(2)
Let
u
(
x
,
t
) =
v
(
x
,
t
)
e
α
t
. Show that by picking the constant
α
cleverly,
v
satisfies equation
(1) with
c
=
0 as well as (2).
Moral: one can easily reduce understanding equations (1)(2) to the special case
c
=
0.
b)
Generalize this to
u
t
+
a
(
t
)
u
=
Δ
u
where
a
(
t
)
is any continuous function by seeking
u
(
x
,
t
) =
ϕ
(
t
)
v
(
x
,
t
)
and picking the function
ϕ
(
t
)
cleverly,
2. In a bounded region
Ω
⊂
R
n
, use the maximum principle to prove a uniqueness theorem for
solutions
u
(
x
,
t
)
of the inhomogeneous equation
u
t

Δ
u
=
F
(
x
,
t
)
in
Ω
with
u
(
x
,
0
) =
f
(
x
)
,
in
Ω
and
u
(
x
,
t
) =
ϕ
(
x
,
t
)
for
x
∈
∂Ω
,
t
≥
0
.
3. Let
Ω
⊂
R
n
be a bounded region with smooth boundary
∂Ω
and let
u
(
x
,
t
)
satisfy the heat
equation
u
t
=
Δ
u
for
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 Spring '09
 Math, Differential Equations, Equations, Partial Differential Equations, Boundary value problem, Boundary conditions, Dirichlet boundary condition, smooth boundary ∂Ω

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