MATH 656 Supplementary Homework
2010
Spring
3. (Telegraph equation) Let Ω ⊂ RN be a bounded open set, A, B be constants,
g ∈ H01 (Ω) and h ∈ L2 (Ω).
a) Show there is a unique suitably defined solution of
utt − ∆u + Aut + Bu = 0
u(x, t) = 0
u(x, 0) = g(x)
ut (x, 0) = h(x)
x∈Ω t>0
x ∈ ∂Ω t > 0
x∈Ω
x∈Ω
by using a change of variables u(x, t) = eσt v(x, t) to get an equivalent problem with
no ut term (and so existence/uniqueness theorems from the text may be applied).
b) If we define the energy functional
Z
2
1
E(t) =
ut (x, t) + |Du|2 (x, t) dx
2 Ω
show directly that E 0 (t) ≤ M E(t) for some constant M . Be explicit about how M
depends on A, B and Ω, and try to find the smallest M that works.
c) Express the solution explicitly in the form
u(x, t) =
∞
X
cn (t)wn (x)
n=1
where {wn }∞
n=1 are the usual Dirichlet eigenfunctions of −∆. If A > 0 is given, for
what range of values of B is the solution exponentially decaying as t → ∞?
4. (Comparison principle) Let Ω ⊂ RN be a bounded open set, f ∈ C 1 (R), and
suppose that
ut − ∆u − f (u) ≥ vt − ∆v − f (v)
u(x, t) ≥ v(x, t)
u(x, 0) ≥ v(x, 0)
(x, t) ∈ QT
(x, t) ∈ ST
x∈Ω
Show that u(x, t) ≥ v(x, t) ∈ QT . (Suggestion: find a linear problem satisfied by
u − v.)