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Necessity of Gevrey-type Levi Conditions for Degenerate Schrodinger Equations

Volume 5, Number 1 (2014), 52 - 70

Necessity of Gevrey-type Levi Conditions for Degenerate Schrodinger Equations

Communicated By: 
Alain Haraux
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Abstract

We consider the Cauchy problem $$ Lu:=i\partial_tu + t^{\ell} \Delta_xu+\sum_{j=1}^nt^k b_j(t,x)\partial_{x_j}u=0,\,\,\,u(0,x)=\varphi(x),$$ with $k \in [0,\ell)$ and some slow decay conditions for the real parts $\Re b_j(t,x)$ of the coefficients of the convection term. We discuss the necessity of Gevrey well-posedness.