32 I. Theory of Distributions

and

u−(t) = lim

(x,t)→(x(t),t)

u(x, t), x x(t).

Let u(x, t) be a distribution solution to a nonlinear equation

∂u

∂t

+

∂

∂x

f(u) = 0,

i.e.,

R2

−u(x, t)

∂ϕ

∂t

− f(u)

∂ϕ

∂x

dxdt = 0

for any ϕ(x, t) ∈ C0

∞(R2).

We assume that f ∈

C∞(R1).

Prove

that the following condition holds:

dx

dt

=

f(u+(t)) − f(u−(t))

u+(t) − u−(t)

.

This condition is called the Rankine-Hugoniot condition.

11. Let f(x1) be the Cantor function on [0, 1], i.e., f(x1) is continuous,

nondecreasing, f(0) = 0, f(1) = 1, and f (x1) = 0 a.e.

Set f0(x1) = f(x1) on [0, 1] and f0(x1) = 0 for x1 that does not

belong to [0, 1].

Prove that the distribution derivative of f0(x1) is the following

functional:

df0

dx

(ϕ) = ϕ(1) −

1

0

ϕ(x1)df(x1),

where

1

0

ϕ(x1)df(x1) is the Stieltjes integral.

12. Prove that there is no distribution on

R1

such that its restriction

to (0, +∞) is a regular functional in D (0, +∞) corresponding to

f(x1) = e

1

x2

1

.