On classification of global dynamics for energycritical equivariant harmonic map heat flows and radial nonlinear heat equation
Abstract
We consider the global dynamics of finite energy solutions to energycritical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices $D\geq3$; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^{1}$bounded radial solutions to (NLH) in dimensions $N\geq7$, building upon soliton resolution for such solutions. To our knowledge, this provides the first rigorous classification of bubble tree dynamics within symmetry. We introduce a new approach based on the energy method that does not rely on maximum principle. The key ingredient of the proof is a monotonicity estimate near any bubble tree configurations, which in turn requires a delicate construction of modified multibubble profiles also.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.04247
 arXiv:
 arXiv:2404.04247
 Bibcode:
 2024arXiv240404247K
 Keywords:

 Mathematics  Analysis of PDEs;
 35K58 (primary);
 35B40;
 37K40;
 58E20
 EPrint:
 44 pages