Cazenave, Thierry; Dickstein, Flávio; Weissler, Fred B. Multi-scale multi-profile global solutions of parabolic equations in \(\mathbb{R}^N \). (English) Zbl 1250.35030 Discrete Contin. Dyn. Syst., Ser. S 5, No. 3, 449-472 (2012). The paper concerns certain concepts that extend the notions of (forward) self-similar and asymptotically self-similar solutions. First, the authors review these fundamental concepts and the basic known results for heat equations on \(\mathbb{R}^N \). Then, the authors examine the possibility that a global solution might not be asymptotically self-similar. Namely, it is shown that the asymptotic form of a solution can evolve differently along different time sequences going to infinity. Indeed, there exist solutions that are asymptotic to infinitely many different self-similar solutions, along different time sequences, all with respect to the same time dependent rescaling. The authors show an explicit relationship between this phenomenon and the spatial asymptotic behavior of the initial value under a related group of dilations. In addition, it is shown that a given solution can exhibit nontrivial asymptotic behavior along different time sequences going to infinity, and with respect to different time dependent rescalings. Reviewer: Vyacheslav I. Maksimov (Ekaterinburg) Cited in 12 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35K05 Heat equation 35K58 Semilinear parabolic equations 35C06 Self-similar solutions to PDEs Keywords:rescaling; decay rate; dilation properties; asymptotically self-similar solutions PDFBibTeX XMLCite \textit{T. Cazenave} et al., Discrete Contin. Dyn. Syst., Ser. S 5, No. 3, 449--472 (2012; Zbl 1250.35030) Full Text: DOI