I would like to know the regularity of the wave equation under simple conditions. Consider the wave equation on the half line: $$ \begin{cases} u_{tt}-\triangle u = 0, \quad (x,t) \in (0, \infty]\times (0, \infty)\\ u(x, 0) = \phi(x), \quad x \in (0, \infty] \\ u_t(x, 0) = \psi(x), \quad x \in (0, \infty] \\ u(0, t) = 0, t\in(0, \infty) \end{cases} $$ where $u:(0, \infty]\times(0, \infty)\to\mathbb{R}$ is unknown. The PDE book by Evans says that the solution $u \notin C^2(\mathbb{R})$, unless $\phi''(0)=0$ [see Evans 2.4.1]. So, what is the regularity of the solution $u$ solving the above equation?
Also, I would like to know some references (books, papers, ...) since I am struggling with funding them.
Thanks you.