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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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1 Answer

up vote 1 down vote accepted

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.

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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
    
@Nik here it is stated that the first example is made by Kovalevskaya and both are mentioned: books.google.ru/… –  Andrew Mar 8 '12 at 15:24
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