u[t_,x_,k_]=Exp[-x^2/4/k/t]/Sqrt[4 Pi k t]

u[1,3,1]

Plot[u[1,x,1],{x,-10,10},PlotRange->{0,0.5}]

Plot[u[5,x,1],{x,-10,10},PlotRange->{0,0.5}]

Animate[
Plot[u[t,x,1],{x,-10,10},PlotRange->{0,0.5}],{t,0,5}]

Plot[u[0.00001,x,1],{x,-0.01,0.01},PlotRange->{0,100}]