For a better animation of the solution from NDSolve

To prevent shaking, try to add ImagePadding and for the other issue, you can fix the vertical plot range.

mpl = Table[
   Plot3D[myfun[x, y, t], {x, 0, L}, {y, 0, L}, 
    PlotRange -> {Automatic, Automatic, {0, 6}}, PlotPoints -> 40, 
    ImageSize -> 400, 
    PlotLabel -> Style["t = " <> ToString[t], Bold, 18], 
    ImagePadding -> 30], {t, 0, 27, 1}];

I only post this as a way (without distortion or dealing with multiple scales)to illustrate the initial Gaussian (flat relative to final range) with MeshFunctions and using ColorFunction. I have voted for Nasser's answer.

fun[t_] := Legended[Show[
   Plot3D[myfun[x, y, t], {x, 0, 20}, {y, 0, 20}, 
    MeshFunctions -> (#3 &), Mesh -> {{0, 1, 1.1, 1.2, 2, 3, 4, 5}}, 
    MeshStyle -> Thick, 
    ColorFunction -> Function[{x, y, z}, ColorData["Rainbow"][z/6]],
    ColorFunctionScaling -> False,
    PlotRange -> {0, 6},
    PlotPoints -> 40,
    PerformanceGoal -> "Quality",
    PlotLabel -> Style[Row[{"t= ", t}], 20, White, Bold]], 
   Background -> Black], BarLegend[{"Rainbow", {0, 6}}]]

The gif was exported from fun/@Range[0,27,0.5] at 8 frames per second (using "DisplayDurations".