This is second order in the time derivative, so there is a utt term. But that should make in not in the realm of pdepe, which allows only ut terms.
Personally, I would just approximate the derivatives using second order finite differences. Convert the entire problem to that of solving a linear system of equations. Given a discretization with nt nodes in t and nx nodes in x, system will be nt*nx by nt*nx. Using sparse to formulate the problem, it will be solvable, though hyperbolic. And that means it may be poorly conditioned, so I would be looking carefully at the singular values and the condition number.
As I said, easily writable. Why not try it? I ask this, because I've never seen a problem as simply stated that was not homework. And if this is homework, I won't do homework. I will help, if you make a credible effort. If you come close enough, I might even show what I would try.
your
