Consider the diffusion equation
$\partial_t u =\partial_{xx} u$
where $x \in [-1,1]$ and $t>0$, subject to IC
$u(x,0)= \frac{1}{1-x}$.
If I construct a power series in $t$ at the origin using the PDE+IC as in Cauchy-Kovaleska theorem (CK) I get
$u(0,t)=\sum_{n=0}^\infty \frac{2n!}{n!} t^n$
which is clearly divergent $\forall t \neq 0$.
My question is why did this happen? And what changes should be made to get a convergent series? CK only requires analicity in a neighbourhood of the origin. And although $\frac{1}{1-x}$ is not analytic, it is analytic in a nbh of $0$. I guess what I am trying to understand with this question is a simple case where CK hypothesis fail and why.
Thank you!