On the admissibility of singular perturbations in Cauchy problems (Q917776)

Consider the singulary perturbed Cauchy problems for Kowalewskian operators with constant coefficients: \[ (\epsilon \cdot P_ 1(D)+P_ 2(D))u(x)=0\text{ in } R^ n;\quad D_ 1^{j-1}u(x)|_{x_ 1=0}=\phi_ j(x'),\quad j=1,...,m. \] Put \(\Phi =(\phi_ 1,...,\phi_ m)\). When the Cauchy problems are uniquely solvable, we can set a problem of the convergence of solutions. We give a necessary and sufficient condition for the convergence of solutions for all \(\Phi\) in \({\mathcal F}^{-1}(C^{\infty}_ 0(B_ 0))^ m\). In particular, if \(ord P_ 1-ord P_ 2\geq 3\) then there exists a data \(\Phi\) such that the analytic solutions diverge.