I am not really sure where to even start, or how one might go about doing a problem such as the one that follows,
Assume $a < 1.$ Prove that the solution to the mixed problem $$ u_{tt} - \Delta u = 0, \text{ in } \{(t,x) \in \mathbb{R} \times \mathbb{R^3}, x_3 > 0\}, \; u_{x_3} - au_t = 0 \text{ on } x_3 = 0,$$ with initial data $u(0,x) = f(x), \; u_t(0,x) = g(x)$ will vanish in the hemisphere $|x| < R - |t|, x_3 > 0$ when $f$ and $g$ vanish for $|x| < R.$ Why doesn't the result hold when $a = 1?$
If anyone could explain a possible method to approach such a problem, hints or first steps, full solutions, anything is appreciated.
Also if anyone knows of a good reference where this type of material is discussed, please feel free to leave a comment citing it. I've been studying for an "applied" PDE qualifying exam and I feel somewhat lost, seeing as all they taught us was the material in the first half of Evans. Thanks in advance.