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The problem is the following:
\begin{align*} &u_{tt}=u_{xx}, \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}_+ \times (0,\infty) \\ &u=g, \ u_t=h, \ \ \ \ \text{on}\ \mathbb{R}_+ \times \{t=0\} \\ &u=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \{x=0\} \times (0,\infty) \end{align*} where $g,h$ are $C^2$ and satisfy $g(0)=h(0)=0$.
Does this problem always have a $C^2$ solution? If yes, is it unique?
In Evans's book (p. 69) he uses the method of reflection to construct a solution, which in order to be $C^2$, more assumptions are needed. This is that solution:
\begin{equation} u(x,t)= \begin{cases} \frac{1}{2}(g(x+t)+g(x-t))+\frac{1}{2}\int_{x-t}^{x+t}h(y)dy & \text{if } x \ge t \ge 0\\ \frac{1}{2}(g(x+t)-g(t-x))+\frac{1}{2}\int_{-x+t}^{x+t}h(y)dy & \text{if } 0\le x \le t \end{cases} \end{equation}
Even if $h=0$, the solution above is not $C^2$, unless $g''(0)=0$.