Well-posedness and smoothing effect for generalized nonlinear Schrödinger equations

Abstract

We improve the result obtained by one of the authors, Bienaimé (2014), and establish the well-posedness of the Cauchy problem for some nonlinear equations of Schrödinger type in the usual Sobolev space $H^s\left({ℝ}^n\right)$ for $s>\frac{n}{2}+2$ instead of $s>\frac{n}{2}+3$. We also improve the smoothing effect of the solution and obtain the optimal exponent.