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Multi-scale multi-profile global solutions of parabolic equations in \(\mathbb{R}^N \). (English) Zbl 1250.35030

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.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
35K58 Semilinear parabolic equations
35C06 Self-similar solutions to PDEs
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