On the weak admissibility of singular perturbations in Cauchy problems (Q923214)

Consider the singularly perturbed Cauchy problems for kowalewskian operators with constant coefficients: \[ (\epsilon \cdot P_ 1(D)+P_ 2(D))u(x)=0\text{ in } {\mathbb{R}}^ n;\quad D_ 1^{j-1}u(x)|_{x_ 1=0}=\phi_ j(x'),\quad j=1,...,m. \] Put \(\Phi '=(\phi_ 1,...,\phi_{m'})\), \(\Phi ''=(\phi_{m'+1},...,\phi_ m)\), and \(\Phi =(\Phi ',\Phi '')\). Here \(m=ord P_ 1>m'=ord P_ 2\). When the Cauchy problems are uniquely solvable, we can set a problem of the convergence of solutions. Let the reduced problem, which is the Cauchy problem for \(\epsilon =0\), be given. Then \(\Phi '\) is given. We give a necessary and sufficient condition that for every \(\Phi '\) in \(F^{-1}(C^{\infty}_ 0(B_ 0))^{m'}\) there exists \(\Phi ''\) in \(F^{-1}(C^{\infty}_ 0(B_ 0))^{m-m'}\) such that the solution converges. This is an algebraic condition on the symbols \(P_ 1\) and \(P_ 2\).