The problem is an exercise of my pde assignment.
Consider for $f,g : \mathbb R \to \mathbb R$ the Cauchyproblem $$u_{tt}(t,x) + u_{xx}(t,x) = 0$$ $$ u(0,x) = f(x) , u_t (0,x) = g(g)$$ Show, that in a Neighbourhood of $0$ there exists a unique analytical solution, if f and g are analytical in a Neighbourhood of $0$.
The D'Alembert fromula gives us the solution $$u(x,y)=\dfrac{f(x+iy)+f(x-iy)}{2}-\dfrac{i}{2}\int_{x-iy}^{x+iy}g(t)~dt$$ but as $f:\mathbb R \to \mathbb R$ this is not defined.
So I considered to solve the equation by seperation of variables, which made me wonder, if this method provides all possible solutions. Or is it possible that there are solutions where $u(t,x)$ can't be sperated.
The next question would be, is the problem is well posed in $\mathcal S (\mathbb R)$ or $H^k(\mathbb R)$.
