Miao, Changxing; Zhang, Bo; Zhang, Xiaoyi Self-similar solutions for nonlinear Schrödinger equations. (English) Zbl 1076.35118 Methods Appl. Anal. 10, No. 1, 119-136 (2003). Summary: We study self-similar solutions for nonlinear Schrödinger equations \[ iu_t+\Delta u= \mu|u|^\alpha u,\quad x\in\mathbb{R}^n,\quad t\geq 0 \] using a scaling technique and the partly contractive mapping method. We establish the small global well-posedness of the Cauchy problem for nonlinear Schrödinger 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 Schrödinger equations with small data admit a class of self-similar solutions. Our results improve and extend well-known results of F. Planchon, T. Cazenave and F. B. Weissler and F. Ribaud and A. Youssfi. Cited in 6 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs Keywords:global well-posedness; Cauchy problem; non-reflexive Banach spaces; a priori; estimates in Besov spaces PDFBibTeX XMLCite \textit{C. Miao} et al., Methods Appl. Anal. 10, No. 1, 119--136 (2003; Zbl 1076.35118) Full Text: DOI