Actually i was just playing around with these equations, checking out how i can use mathematica to gain a deeper understanding, so my problem isn't really clearly defined. I didn't expect the solution would be so complicated though. Polar coordinates seem to be the way to go, i will try my hand on that.

Solve 2D Wave equation with DSolve?

sorry. hit the wrong key. discard this

See

Partial Differential Equations With Mathematica (Physics Series) (Englisch) Gebundene Ausgabe – Januar 1993

von Dimitri Vvedensky (Autor)

https://www.amazon.de/Partial-Differential-Equations-Mathematica-Physics/dp/0201544091/ref=sr_1_1?ie=UTF8&qid=1481669125&sr=8-1&keywords=vvedensky+partial

and: is your problem confined to a certain region? your ic seems to show that you could look for solutions on a (perhaps inifinite) disk. then you should try polar coordinates. I think in this case the radial solutions are Bessel functions and sums of Bessel functions. Perhaps you can develop the Exp[ - r^2 ] in a series of Bessels and then you have a solution with your ic.

That would not be convenient due to the series, but......

I tried it again with "appropriate" numbers of ic's and bc's, but without success.

Perhaps you should tackle your problem with the old fashioned method of a product Ansatz

u[ x, y, t ] =f1[ x ] f2 [y ] f3 [ t ]

Sounds like progress. I'll play around a bit more when I get home from work.

I didn't have much luck with DSolve, but I did with NDSolve. Please have a look at the boundary and initial conditions. You should have 2 boundary conditions for x, 2 boundary conditions for y, and 2 initial condition equations for t. I made some up. These could be other things.

Clear[L, c, weq2, bc2, ic2, sol2, u, x, y, t, m, n]
weq2 = Laplacian[u[x, y, t], {x, y}] == D[u[x, y, t], {t, 2}]

bc2 = {u[x, Pi, t] == 0, u[x, 0, t] == 0, u[0, y, t] == 0, 
  u[Pi, y, t] == 0}
ic2 = {u[x, y, 0] == Sin[m x] Sin[n y], 
  Derivative[0, 0, 1][u][x, y, 0] == 0}

sol2 = DSolve[{weq2, bc2, ic2}, u[x, y, t], {x, y, t}, 
  Assumptions -> {Element[m | n , Integers]} ]

DSolve failed to find a solution. I had no trouble doing the same for 1D, but 2D it just isn't happy. Maybe someone can figure that out. So on with NDSolve

nsol2 = NDSolve[{weq2, bc2, ic2} /. {m -> 2, n -> 2}, 
  u, {x, 0, Pi}, {y, 0, Pi}, {t, 0, 5}]

ListAnimate[
 Table[Plot3D[u[x, y, t] /. nsol2, {x, 0, \[Pi]}, {y, 0, \[Pi]}, 
   AxesLabel -> Automatic, 
   PlotRange -> {{0, \[Pi]}, {0, \[Pi]}, {-1, 1}}], {t, 0, 5}]]

You should get an animation with time ....