Based on some older Math group thread my problem is the following (0 < t):

F:= (x,t) -> Int(exp(-t*eta^2+x*eta)/(1+exp(eta)),eta = -infinity .. infinity);

satisfies 0 = 'diff(F(x,t),t) + diff(F(x,t),x$2)' and for that PDE Maple gives

  pdsolve(PDE, f(x,t),build): combine(%):
  subs(_c[1]=c,_C1=c1,_C2=c2,_C3=c3,%): rhs(%);
  S:=unapply(%, x,t);

    S := (x, t) -> c3*c1*exp(c^(1/2)*x-c*t)+c3*c2*exp(-c^(1/2)*x-c*t)

by separation of variables.

I am interested in t=1/2 ( to get (F(x,1/2) ) and for that define

G:= x -> Int(exp(-1/2*(eta-x)^2)/(1+exp(eta)),eta = -infinity .. infinity);

Then we have 'exp(-1/2*(x)^2) * F(x,1/2) = G(x)'; # combine(%): is(%); # =true

Being a bit lame I do not really analytically determine G(+- infinity), but
by plotting and taking large values (say at x=+-40), which gives me 0
and sqrt(2*pi) for x=-infinity. So for very small x that G(x) *not* zero.

However using the solution S one gets 0 at both ends as a sum of 2 Gaussians:

  'exp(-1/2*x^2)*S(x,1/2)';
                                2
                               x
                        exp(- ----) S(x, 1/2)
                               2

  expand(%): combine(%): combine(%,exp): 
  completesquare(%,x): simplify(%,size);

             /               1/2 2                   1/2 2 \
             |         (x - c   )              (x + c   )  |
          c3 |c1 exp(- -----------) + c2 exp(- -----------)|
             \              2                       2      /

I can not find my fault :-(

www.mapleprimes.com/files/102_heat_problem.mws