Essentially I have a second order PDE for x[t,z], and I define x[t=0,z] and x[t,z=0]. As far as I understand this is sufficient to solve this equation, or do I need more boundaries? Is my syntax wrong?
I confirm the unreasonable message of Mathematica reported by Ilia. First, there is no discrepancy between the initial and the boundary conditions, and they are given according to the NDSolve rules.
I also checked, if Mathematica did not got "scared" by all DiracDeltas and HeavisideTheta functions. To this end I replaced these functions by more trivial ones that in no case may cause Mathematica problems: