I want to know if there exists any solution of the equation
$$\eqalign{ & {u_t} = - {u_{xx}} \cr & u(0,1) = u(0,\pi ) = 0 \cr & u(0,x) = {u_0}(x) \cr} $$
We can easily obtain the solution of this equation by using variables separation method $$u(t,x) = \sum\limits_{n \geqslant 1} {{c_n}{e^{{n^2}t}}\sin (nx)} $$ but this solution is not defined for any $t>0$, the serie diverges.
I'm write? Thanks.