1. Using the method of characteristics, solve the following initial problems:

(a)

u

t

(t, x) + (1 − 2u(t, x))u

x

(t, x) = 0,

t > 0, x ∈ R

u(0, x) = χ

[0,1]

(x),

x ∈ R.

(b)

u

t

(t, x) + u(t, x)u

x

(t, x) = 1,

t > 0, x ∈ R

u(0, x) = −2x.

2. Compute the weak derivatives of the following distributions:

(a) e

|x|

(b) | sin x|

(c) f (x) =



|x|

x ≤ 1

x − 1

x > 1.

(d) δ

0

− δ

1

3. Let f (x) =

1

√

x

χ

[1,∞)

(x). For which p ≥ 1 do we have f ∈ L

p

(R)?

4. Show that for a ∈ R, kaf k

L

p

(0,1)

= |a|kf k

L

p

(0,1)

.

5. Let 1 ≤ r ≤ p. Show that k|f |

r

k

L

p
r

(0,1)

= kf k

r
L

p

(0,1)

.

6. Show that e|

x−

1
2

| ∈ H

1

(0, 1).

7. Using the Rankine-Hugoniot condition, solve the Burgers’ equation with initial data

u

0

(x) =







1,

x < 0

1 − x,

0 ≤ x ≤ 1

0,

x > 0

(i.e. compute the solution for all x ∈ R and t > 0).

8. Write the weak (i.e. Sobolev-space) formulation of the following problem

−u

xx

(x) + u(x) = f (x)

u(0) = u(1) = 0.

9. Write the weak formulation of the problem

−u

xx

(t, x) = f (t, x)

u

x

(0) = a

u

x

(1) = b

(with a, b ∈ R).