It's not really a PDE. You have no derivatives in s.
Here is a way one might go about getting say the half where t<=s. I'll make up some functions so it will look a bit like the original question.
r3D[t_?NumberQ, s_?NumberQ] := Cos[t^2 + s]
ysol[t_?NumberQ] := Sin[t]
We'll make $s$ an index parameter. For a given set of values we'll solve an ODE in $t$.
eqns[s_?NumberQ] := {D[
y[s][t], {t, 2}] == (y[s][t] - r3D[t, s]*D[y[s][t], t]),
y[s][0] == ysol[s], y[s][s] == 0};
Now we make a table of solutions.
soln = Flatten[
Table[NDSolve[eqns[s], y[s][t], {t, 0, s}], {s, .05, 1., .05}]];
Make a rectangular list of values. Set to 0 outside the range of our solutions. One can check that the boundary conditions were respected, up to a smallish factor of $MachineEpsilon.
vals = Table[
If[t >= s, 0, y[s][t] /. soln], {s, .05, 1., .05}, {t, 0, 1, .05}];
InterpolatingFunction::dmval: Input value {0.1} lies outside the range of data in the interpolating function. Extrapolation will be used. >>
InterpolatingFunction::dmval: Input value {0.1} lies outside the range of data in the interpolating function. Extrapolation will be used. >>
InterpolatingFunction::dmval: Input value {0.15} lies outside the range of data in the interpolating function. Extrapolation will be used. >>
General::stop: Further output of InterpolatingFunction::dmval will be suppressed during this calculation. >>
I'm too tired to understand where/why I am outside the interpolation boundaries. Apologies. Anyway, we get a perhaps plausible set of values.
ListPlot3D[vals]

p3D
etc very long functions, or could they be included in the question. That might help people give better answers. $\endgroup$ – tkott Mar 4 '12 at 16:32t=0
the conditiony[t,0]==ysol[t]
might be a contradition unlessysol[0]==0
. It might help to set the region to something like{t, 0, 1}, {s, 0, t}
and then develop the solution in two parts. $\endgroup$ – Matariki Mar 5 '12 at 1:42