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Positive solutions for critical elliptic systems in noncontractible domains
Local wellposedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$
1.  Department of Mathematics, Kyoto University, Kyoto 6068502, Japan 
$ iu_t+u_{x x}=\bar u^2$
in $H^s(\mathbb R)$ for $s\ge 1$ and illposedness below $H^{1}$. The same result for another quadratic nonlinearity $u^2$ was given by I. Bejenaru and T. Tao, Sharp wellposedness and illposedness results for a quadratic nonlinear Schrödinger equation, J. Funct. Anal. 233 (2006), but the function space of solutions depended heavily on the special property of the nonlinearity $u^2$. We construct the solution space suitable for the nonlinearity $\bar u^2$.
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