Gain of regularity for semilinear Schrödinger equations
The author considers the Cauchy problem for a semilinear Schrödinger equation with right hand side of the form \(f(u,\partial_xu)\). It is well known that in this case the so-called loss of derivatives occurs. A local in time \(t\) existence and uniqueness result is proved, if \(f(u,v)\) has a quadratic growth near \((u,v)=0\) and the initial data \(u_0\) belongs to some weighted in \(x\) Sobolev class \(H^{m,l}\). A local existence and uniqueness result is also proved for \(f(u,v)\) having a cubic nonlinearity near \((u,v)=0\) and for Cauchy data \(u_0\) belonging to some classical Sobolev space \(H^m\). The last theorem deals with gain of regularity of the solution of the problem under consideration. To do this the author assumes that \(f(u,v)\) has a cubic nonlinearity and is gauge invariant while \(u_0\in H^{m,m_1}\), \(m_1>0\), \(m_1\in \mathbb{Z}\).
scientific article; zbMATH DE number 1395777