a = 2; b = 1;

t =.

mm  =  nn  =  10;

λ[m_, 
        n_]  := (π^2) (((m^2)/(a^2))  +  ((n^2)/(b^2)))

f[x_]  :=  Which[x  <   .5, x, True, (2 - x)/3]

g[x_]  :=  Which[x  <   .3, x, True, (1 - x) 3/7]

cx // ClearAll; cy // ClearAll; c1 // ClearAll;

cx[m_]  :=  cx[m]  =  (2/a) 
          NIntegrate[
            f[ξ] Sin[((m π ξ)/a)] , {ξ, 0, a}];  
    cy[m_]  :=  cy[m]  =  (2/b) 
          NIntegrate[
            g[ξ] Sin[((m π ξ)/b)] , {ξ, 0, b}];  
    c1[m_, n_]  :=  c1[m, n]  =  cx[m] cy[n];

F1[x_, y_, 0]  =  f[x] g[y];

F1[x_, y_, t_]  =  
        Sum[c1[m, 
              n] Exp[- λ[m, 
                    n] t] Sin[((m π x)/a)] Sin[((n π y)/b)], {m, mm}, {n, nn} ];

f[x_]  :=  Which[x  < 1.5, x, True, (2 - x) 3]

g[x_]  :=  Which[x  <   .7, x, True, (1 - x) 7/3]

cx // ClearAll; cy // ClearAll; c2 // ClearAll;

cx[m_]  :=  cx[m]  =  (2/a) 
          NIntegrate[
            f[ξ] Sin[((m π ξ)/a)] , {ξ, 0, a}];  
    cy[m_]  :=  cy[m]  =  (2/b) 
          NIntegrate[
            g[ξ] Sin[((m π ξ)/b)] , {ξ, 0, b}];  
    c2[m_, n_]  :=  c2[m, n]  =  cx[m] cy[n];

F2[x_, y_, 0]  =  f[x] g[y];

F2[x_, y_, t_]  =  
        Sum[c2[m, 
              n] Exp[- λ[m, 
                    n] t] Sin[((m π x)/a)] Sin[((n π y)/b)], {m, mm}, {n, nn} ];

F[x_, y_, t_]  =  F1[x, y, t]  -   .08 F2[x, y, t];

dt  =   .002;  kk  =  50; 
    Do[tt  =  k dt; 
      Plot3D[F[x, y, tt], {x, 0, a}, {y, 0, b}, Boxed -> False, 
        Axes -> False, Mesh  ->  False, 
        PlotPoints  -> 50 {1, 1}, 
        PlotRange  ->  .14 {- .3, 1}, 
        ViewPoint -> {1.3, 2.4,  .6}, 
        BoxRatios  ->  {2, 1, 1}], {k, 0, kk}]