Consider the homogeneous heat system with Dirichlet B.C
\begin{array}{c} u_{t}-u_{xx}=0 \\ u(0,t)=u(\pi ,t)=0 \\ u(x,0)=f(x).% \end{array} It is well known that the solution of the system above is given by
$$u(t,x)=\sum_{n\geq 1}A_{n}e^{-n^{2}t}\sin (nx)$$,
Also, it is well known that $u\in C^{\infty }((0,T]\times (0,\pi ))$ for any $f\in L^{2}(0,\pi ).$ (Smoothness effect).
My question is : What if there exists some function $f\in X$ such that $% A_{n}\sim e^{n^{2}},$ where $X$ is some Hilbert or Banach space, the solution will not exist,
Is there any initial condition and some $X$ which satisfy that ?
Thanks.