@terrytao The Lewy example may be eligible to a CK treatment, it has terrible pathologies: there exists a second category (Baire sense) subset of $\C^\infty$ rhs $f$ such that the equation $(Lewy) u=f$ has no distribution solutions, even locally. Moreover the CK solutions are extremely unstable: take an analytic data $u_0$ on some analytic hypersurface. Then in any $\C^\infty$ neighborhood of $u_0$, there exists a $C^\infty$ function $w_0$ such that the IVP has no solutions. Best, Bazin.