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Self-similar solutions for nonlinear Schrödinger equations. (English) Zbl 1076.35118

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.

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
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