Interesting.

I don't get an error message, but neither a solution.....

DSolve[{-
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "2"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, t] + 
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, t] + 
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, t] == 0, 
  u[x, y, 0] == E^(-x^2 - y^2), 
\!\(\*SuperscriptBox["u", 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, 0] == 0, u[x, 0, 0] == 0, 
  u[0, y, 0] == 0}, u, {x, y, t}]

Thanks for the replys! NDSolve is really interesting and the animations are great, but i would also really like to take a look at the analytical solutions for these equations. I tried some ic's (plane waves,gaussians,..) in 2D, now with the 4 bc's, but everytime i just got my input as output, without an error message (as with your codes).I've found something in the DSolve documentation though, under Scope->Hyperbolic PDEs, quote:

Dirichlet problem for the wave equation in a rectangle:

weqn = D[u[x, y, t], {t, 2}] == Laplacian[u[x, y, t], {x, y}];
ic = {u[x, y, 0] == (1/10) (x - x^2) (2 y - y^2), 
   Derivative[0, 0, 1][u][x, y, 0] == 0};
bc = {u[x, 0, t] == 0,
    u[0, y, t] == 0, u[1, y, t] == 0, u[x, 2, t] == 0};

The solution is a doubly infinite trigonometric series:

(sol = FullSimplify[
    u[x, y, t] /. DSolve[{weqn, ic, bc}, u, {x, y, t}][[1]], 
    K[1] \[Element] Integers && 
     K[2] \[Element] Integers]) // TraditionalForm
\!\(\*
UnderoverscriptBox[
StyleBox["\[Sum]", "InactiveTraditional"], 
RowBox[{
TemplateArgBox[
RowBox[{"K", "[", "1", "]"}]], "=", 
TemplateArgBox["1"]}], 
TemplateArgBox["\[Infinity]"]]\(\*
UnderoverscriptBox[
StyleBox["\[Sum]", "InactiveTraditional"], 
RowBox[{
TemplateArgBox[
RowBox[{"K", "[", "2", "]"}]], "=", 
TemplateArgBox["1"]}], 
TemplateArgBox["\[Infinity]"]]\*
TemplateArgBox[
FractionBox[
RowBox[{"32", " ", 
RowBox[{"(", 
RowBox[{
SuperscriptBox[
RowBox[{"(", 
RowBox[{"-", "1"}], ")"}], 
RowBox[{"K", "[", "1", "]"}]], "-", "1"}], ")"}], " ", 
RowBox[{"(", 
RowBox[{
SuperscriptBox[
RowBox[{"(", 
RowBox[{"-", "1"}], ")"}], 
RowBox[{"K", "[", "2", "]"}]], "-", "1"}], ")"}], " ", 
RowBox[{"cos", "(", 
RowBox[{"\[Pi]", " ", "t", " ", 
SqrtBox[
RowBox[{
SuperscriptBox[
RowBox[{"K", "[", "1", "]"}], "2"], "+", 
FractionBox[
SuperscriptBox[
RowBox[{"K", "[", "2", "]"}], "2"], "4"]}]]}], ")"}], " ", 
RowBox[{"sin", "(", 
RowBox[{"\[Pi]", " ", "x", " ", 
RowBox[{"K", "[", "1", "]"}]}], ")"}], " ", 
RowBox[{"sin", "(", 
RowBox[{
FractionBox["1", "2"], " ", "\[Pi]", " ", "y", " ", 
RowBox[{"K", "[", "2", "]"}]}], ")"}]}], 
RowBox[{"5", " ", 
SuperscriptBox["\[Pi]", "6"], " ", 
SuperscriptBox[
RowBox[{"K", "[", "1", "]"}], "3"], " ", 
SuperscriptBox[
RowBox[{"K", "[", "2", "]"}], "3"]}]]]\)\)

End of quote. Why does this work and the others don't? The only difference i can see are the initial conditions?

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

So I played around. Yeah it seems DSolve recognized something with the ICs that it liked. So that isn't helping you much. However, this might help you with some tricks related to the infinite series. Also notice I used DSolveValue to avoid getting back a rule.

weqn = D[u[x, y, t], {t, 2}] == Laplacian[u[x, y, t], {x, y}];
ic = {u[x, y, 0] == (1/10) (x - x^2) (2 y - y^2), 
   Derivative[0, 0, 1][u][x, y, 0] == 0};
bc = {u[x, 0, t] == 0, u[0, y, t] == 0, u[1, y, t] == 0, 
   u[x, 2, t] == 0};
dsol = DSolveValue[{weqn, ic, bc}, u, {x, y, t}];
ListAnimate[
 Table[Plot3D[
   Activate[dsol[x, y, t] /. Infinity -> 5], {x, 0, 1}, {y, 0, 2}, 
   PlotRange -> {{0, 1}, {0, 2}, {-0.05, 0.05}}], {t, 0, 10, 0.25}]]

Note the Activate function and the replacement of infinity.

I realize you want a nice compact equation, but you might not get it.

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......

Here is another attempt to come nearer to what you want.

Define the wave equation in polar coordinates

DGL[f_] :=  D[f, r, r] + 1/r D[f, r] + 1/r^2 D[f, phi, phi] - D[f, t, t]

A solution is given by

u[r_, phi_, t_, n_, m_] := 
BesselJ[n,BesselJZero[n, m] r] (\[Alpha] Cos[n phi] + \[Beta] Sin[n phi]) (a Cos[ BesselJZero[n, m] t] - b Sin[BesselJZero[n, m] t])

See

DGL[u[r, phi, t, n, m]] // FullSimplify`

Now we try to approximate the e-function by a series of u's. Taking into account that Exp[ - r^2 ] is not dependent on phi we chose n = 0 and calculate the coefficients in the series according to

c[n_] := \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]\(r\ Exp[\(-r^2\)]\ \
BesselJ[0, r\ BesselJZero[0, n]] \[DifferentialD]r\)\)/\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]\(r\ 
SuperscriptBox[\(BesselJ[0, 
     r\ BesselJZero[0, n]]\), \(2\)] \[DifferentialD]r\)\)

Unfortunately the integral in the numerator must be evaluated numerically. But having done this we get something that could be useful.....

Attachments: