# Divergent series solution to PDE - Failure of Cauchy-Kovaleska

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!

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Well, that's actually the example which Kovalevskaya herself gave to demonstrate the neccesity of one of the conditions in the theorem. Namely, the order of differentiation for the variable $t$ in the lhs should be greater or equal to the order of the rhs differential operator.
Thank you. Actually the example she gave was different (although almost the same). She used $u(0,x)=(x^2+1)^{-1}$, at least that is what wikipedia says... – chango Mar 8 '12 at 14:03