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Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations. (English) Zbl 0916.35109
The authors study global solution, including self-similar solutions of the initial value problem for nonlinear Schrödinger equations. The methods are also applied to the nonlinear heat equation. The proof of the existence of the global solutions is obtained by using norms analogous to those used by Cannone and Planchon for the Navier-Stokes systems with Besov norms. It is analyzed the action of the linear Schrödinger group on homogeneous functions. That allows to construct self-similar solutions and also a class of classical $$H^1$$ solutions that are asymptotically self-similar as $$t\to\infty$$. Also, the authors obtain solutions on $$(-\infty,0)$$ which have an asymptotically self-similar blow up behavior at $$t=0$$.