And here is a 2D-example. It runs (at least on my machine) quite long

dd = .1;
cc = 1;
ee = 25.;
f1 = f[x, y, t];
sol = NDSolve[
    {
     D[f1, t] == 
      dd ( D[f1, x, x] + D[f1, y, y]) - cc Exp[-ee (x^2 + y^2)], 
     D[f1, x] == 0 /. x -> -1,
     D[f1, x] == 0 /. x -> 1,
     D[f1, y] == 0 /. y -> -1,
     D[f1, y] == 0 /. y -> 1,
     f[x, y, 0] == 1
     },
    f,
    {x, -1, 1}, {y, -1, 1},
    {t, 0, 15}][[1, 1]];
ff = f /. sol;
Manipulate[
 Plot3D[ff[x, y, t], {x, -1, 1}, {y, -1, 1}, PlotRange -> {0, 1.1}, 
  PlotStyle -> Opacity[.5]], {t, 0, 15}]

Hello Gonzalo. What do you want to describe? I assume a square basin of water with a sink in the middle?

Here is a one-dimensional attempt as a first step:

dd = .1;
cc = 1;
ee = 25.;
sol = NDSolve[
    {
     D[f[x, t], t] == dd D[f[x, t], x, x] - cc Exp[-ee x^2],
     D[f[x, t], x] == 0 /. x -> -1,
     D[f[x, t], x] == 0 /. x -> 1,
     f[x, 0] == 1
     },
    f,
    {x, -1, 1},
    {t, 0, 15}
    ][[1, 1]];
ff = f /. sol;
Manipulate[
 Plot[
  {1 - cc Exp[-ee x^2], ff[x, t]}, {x, -1, 1}, PlotRange -> {0, 1.1}],
 {t, 0, 15}
 ]