Problem - Consider the inviscid Burgers equation: u_t + uu_x = 0

where u: R → R

1. The Cauchy theory: For any u₀ ∈ C¹(R) such that inf_x∈R u_x ≠ -∞, show that there exists a unique classical solution u ∈ C¹([0, T) x R) up to time

T = - (1/inf_x∈R(u₀)_x ≠ +∞) (T = -∞ if inf_x∈R(u₀)x ≥ 0).

2. Blow-up criterion: Show that for any u₀ ∈ C¹(R) such that -∞ < inf_x∈R(u₀)_x < 0 and such that this infimum is attained at a point x₀, we have for all ∈ > 0,

Sup_{(t,x)∈[T -∈, T) x [x0 +Tu0(x0) -∈, x0 + Tu0(x0) + ∈]} |u_x(t, x)| = +∞,

3. Existence of a. profile at blow-up time: Let u₀ ∈ C¹(R) such that - ∞ < inf_x∈R(u₀)x < 0 and such that this infimum is attained on a discrete set of points. Assume in addition that

∀x ∈ R, inf_y→±∞ 1/y-x _x∫^yu_x(z)dz > - 1/T                    (1)

(note that one has ≥ from the definition of T). Show that u(T, x) := lim_{(x', t)→(x, T)} u(x, t) is well defined, continuous on [0, T] x R and C¹ on

([0, T] x R) \S,  S:= {(T, z), z = y + Tu₀(y), u_x(y) = inf_x∈R u_x}.

4. As a counter example of the above limit, take an initial data at T = 1 given by

u₀(x) = - 1/x - x.

Using this, show that there exists a solution such that (1) does not hold and such that the above limit u(T, x) is not defined.