Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic\*.


Is it possible a solution $f$, which is non-analytic at any point of the domain?




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\*Even if the initial conditions are analytic, it is possible that the solution is not [[1](http://www.sciencedirect.com/science/article/pii/S0747717103000178)].



Analyticity of the solutions of PDE
Points off for the rude response to Otis' sincere attempt to help you! 


Deanne is correct, but you can see this simply even with the operator $\partial\_{x\_1}$ in $\mathbb R^n$. If $u(0, x\_2, \ldots, x\_n)$ is nowhere analytic, then the
solution of $\partial u/ \partial\_{x\_1} = 0$ is constant in $x\_1$ but never analytic in the orthogonal directions.


On the other hand, there is a class of operators known as analytic hypoelliptic (which includes both the Laplacian for any analytic metric and the associated heat operator) for which such a statement is true: $L u = f$ and $f$ analytic implies $u$ analytic. 


However as the paper you reference indicates, all sorts of wild things can happen in general; that paper gives an example of a constant coefficient system and analytic ``initial data'' where the solution is not analytic. Actually, that is a rather nonstandard sort of initial value problem where Cauchy data for one component is posed on one hypersurface and Cauchy data for the second component is posed on a transverse hypersurface. 


