Mathematics · Partial Differential Equations

Let u = u(x, t) be the solution of the following initial value problem: ut + 2024ux = 0, x ∈ ℝ, t > 0 u(x, 0) = u0(x), x ∈ ℝ where u0 : ℝ → ℝ is an arbitrary C1 function. Consider the following statements: S1: If At = {x ∈ ℝ : u(x, t) 0. S2: If u0 is Lebesgue integrable, then for every t > 0, the function x → u(x, t) is Lebesgue integrable.