Multi-scale multi-profile global solutions of parabolic equations in RN

Abstract

This paper explores certain concepts which extend the notions of (forward) self-similar and asymptotically self-similar solutions. A self-similar solution of an evolution equation has the property of being invariant with respect to a certain group of space-time dilations. An asymptotically selfsimilar solution approaches (in an appropriate sense) a self-similar solution to first order approximation for large time. Such solutions have a definite longtime asymptotic behavior, with respect to a specific time dependent spatial rescaling. After reviewing these fundamental concepts and the basic known results for heat equations on RN, we examine the possibility that a global solution might not be asymptotically self-similar. More precisely, we show that the asymptotic form of a solution can evolve differently along di erent time sequences going to infinity. Indeed, there exist solutions which are asymptotic to infinitely many different self-similar solutions, along different time sequences, all with respect to the same time dependent rescaling. We exhibit an explicit relationship between this phenomenon and the spatial asymptotic behavior of the initial value under a related group of dilations. In addition, we show that a given solution can exhibit nontrivial asymptotic behavior along different time sequences going to infinity, and with respect to different time dependent rescalings.