<< RealTime3D`

e[x_, a_]  :=  (E^(-(x/a)^2))

fa[x_]  :=  e[x - x0,  .2] ;

x0 = 0. ; 
    u[t_, x_]  := 
      Which[t  <   .0001, fa[x], x  ≤  -1/t, 0, True,  
        fa[(x/(1 + t x))]]

Plot3D[u[t, x], {x, - .5, 2}, {t, 0, 3}, 
      BoxRatios  -> {1, 1,  .2}, Boxed -> False, 
      Axes -> False, Mesh -> False, 
      PlotPoints -> 150]

pp[t_]  :=  
      Plot[u[t, x], {x, - .5, 2}, 
        PlotRange  -> {{- .5, 2}, {- .2, 1.2}}, 
        AspectRatio  ->   .3, 
        PlotStyle  ->  {Thickness[ .004], Hue[ .45]}]

δt =  .05;pp /@ 
      Range[0, 12, δt];