Analytic heat data need not give a time-analytic germ

For $\mathrm{\mid }x\mathrm{\mid }<1$, the geometric identity gives $1\mathrm{/}(1+x^2)={\sum }_{k\ge 0}(-1{)}^k{x}^{2k}$, hence the data are analytic with $g^{(2k)}(0)=(-1{)}^k(2k)!$. If u were analytic, repeated differentiation of its equation and F1 would give ${\mathrm{\partial }}_t^{k}u={\mathrm{\partial }}_x^{2k}u$, beginning at k=0 and using ${\mathrm{\partial }}_t{\mathrm{\partial }}_x^{2k}u={\mathrm{\partial }}_x^{2k}{u}_t={\mathrm{\partial }}_x^{2k+2}u$ at each step. Evaluation on the initial trace gives the asserted derivatives.

The time coefficient is $a_k=(-1{)}^k(2k)!\mathrm{/}k!$. For every fixed t nonzero, $\mathrm{\mid }{a}_{k+1}{t}^{k+1}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{a}_k{t}^k\mathrm{\mid }=2(2k+1)\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$. Thus these terms eventually increase by a factor at least two and do not tend to zero. The time series diverges at every t nonzero and cannot represent an analytic germ. The PDE has total order two, with no u_tt term; its principal symbol is ${\xi }_x^2$ and vanishes at dt. It therefore lies outside the noncharacteristic CK hypothesis.