Methods and Applications of Analysis

Volume 10 (2003)

Number 1

Self-Similar Solutions for Nonlinear Schrödinger Equations

Pages: 119 – 136

DOI: http://dx.doi.org/10.4310/MAA.2003.v10.n1.a7

Authors

Changxing Miao

Bo Zhang

Xiaoyi Zhang

Abstract

In this paper we study self-similar solutions for nonlinear Schrödinger equations using a scaling technique and the partly contractive mapping method. We establish the small global well-posedness of the Cauchy problem for nonlinear Schrodinger equations in some non-reflexive Banach spaces which contain many homogeneous functions. This we do by establishing some a priori nonlinear estimates in Besov spaces, employing the mean difference characterization and multiplication in Besov spaces. These new global solutions to nonlinear Schrodinger equations with small data admit a class of self-similar solutions. Our results improve and extend the well-known results of Planchon [18], Cazenave and Weissler [4, 5] and Ribaud and Youssfi [20].

Full Text (PDF format)