Over the real line, I know that for initial data $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$ s.t. $f,g\in L^2(\mathbb{R})$ [name this (1)] we have uniqueness. My question is:
Without assuming (1), do we still have uniqueness? If not, can you give me a counterexample? In this setting, does D'Alembert's solution and that one obtained by separation of variables (Fourier method) coincide?
Also, in higher dimensions:
On the existence in a bounded open subset $A\subset \mathbb{R}^n$ with regular $\partial A$: Is there any guarantee of the existence of solutions? If not, under what conditions on $A,f,g$ and boundary conditions do we guarantee existence?
On the uniqueness over $\mathbb{R}^n$: What assumptions must be made in order to guarantee uniqueness to the wave equation solution over the whole space? Is there any? If not, could you give me a counterexample or an intuitive explanation?
Thanks in advance.