1.2. Examples of propagation of singularities using progressing waves 15

The singularities are computed by the method of progressing waves. For

n ∈ N, introduce

(1.2.7) hn(x) :=

xn/n!

for x ≥ 0,

0 for x ≤ 0.

Then

(1.2.8)

d

dx

hn+1 = hn , for n ≥ 0 .

Exercise 1.2.4. Show that there are uniquely determined functions an(t)

satisfying

a0(0) = 1/2 and an(0) = 0 for n ≥ 1 ,

and so that for all N ≥ 2,

(1.2.9) ∂t

2

− ∂x

2

+ 1

N

n=0

an(t) hn(x − t) ∈

CN−2(R2)

.

In this case, we say that the series

∞

n=0

an(t) hn(t − x)

is a formal solution of (∂t

2

− ∂x

2

+ 1)u ∈

C∞.

Hints. Pay special attention

to the most singular term(s). In particular show that, ∂ta0 = 0.

Exercise 1.2.5. Suppose that u is the fundamental solution of the Klein–

Gordon equation and M ≥ 0. Find a distribution wM such that u − wM ∈

CM (R2).

Show that the fundamental solution of the wave equation and

that of the Klein–Gordon equation differ by a Lipschitz continuous function.

Show that the singular supports of the two fundamental solutions are equal.

Hint. Add (1.2.9) to its spatial reflection and choose initial values for the

two solutions to match the initial data.

Exercise 1.2.6. Study the fundamental solution for the dissipative wave

equation

(1.2.10) utt − uxx + 2ut = 0 .

Use Theorem 1.1.4 to show that the singular support is contained in the

characteristics through (0, 0). Show that it is not a continuous perturbation

of the fundamental solution of the wave equation. Hint. Find solutions of

(∂t

2

− ∂x

2

+ 2∂t)u ∈

C∞

of the form

∑

n

bn(t) hn(t − x) as in Exercises 1.2.4

and 1.2.5. Use two such solutions as in Exercise 1.2.5.

The method of progressing wave expansions from these examples is dis-

cussed in more generality in chapter 6 of Courant and Hilbert Vol. 2, and in

[Lax, 2006]. The higher dimensional analogue of these solutions are singular