Let u = u(x, t) be the solution of the following initial value problem
\(\rm \left\{\begin{matrix}u_t+2024u_x=0,&x ∈ R, t>0\\\ u(x, 0)=u_0(x), &x ∈ R\end{matrix}\right.\)
where u0 : ℝ → ℝ s an arbitrary C1 function. Consider the following statements:
S1 : If At = {x ∈ ℝ : u(x, t) < 1} and |At| denotes the Lebesgue measure of A, for every t ≥ 0, then |At| = |A0|, ∀t > 0
S2 : If u0 is Lebesgue integrable, then for every t > 0, the function x → u(x, t) is Lebesgue integrable.
1
both S1 and S2 are true
2
S1 is true but S2 is false
3
S2 is true but S1 is false
4
both S1 and S2 are false