So, my problem is the following:
I have a classical IVP (Cauchy problem) for a wave equation on the real line:
$u_{tt} = a^2u_{xx} + f(t,x)$
$u|_{t=0} = u_0(x)$
$u_t|_{t=0} = u_1(x)$
By definition, the IVP is searching for $u\in(C^2(t>0)\cap C^1(t\geqslant0))$.
I want to prove a stronger thing:
I want to prove that if $f(x,y) \equiv 0$ and the solution to the IVP exists (but is not necessarily unique), then:
$u \in C^2(t\geqslant0)$
$u_0 \in C^2(R)$
$u_1 \in C^1(R)$
My initial desire was to try and use the D'Alembert formula and try to differentiate the solution at $t=0$. But the very usage of D'Alembert formula requires at least some constraints on $u_0$ and $u_1$.
I don't even have a clue how to approach such problems. Most of the theorems in the PDE course deal with sufficient conditions, whereas here I need a necessary condition.
I could say something like:
Suppose this is not true, and $lim_{t\to0} u(t,x) =g(x) \neq u_0(x)$
We could differentiate them (since $u\in C^1(t=0)$):
$g_x(x) =? u_{0x}(x)$
Does it tell us anything? Well, we could do something like $g_{xx} = u_{0xx}$, but alas, we are not sure that we can differentiate the second time.
So, I am confused, any suggestions appreciated.