Rigidity results in multi-bubble dynamics for non-radial energy-critical heat equation
Kihyun Kim, Frank Merle
TL;DR
This work establishes, for the energy-critical nonlinear heat equation in dimensions $N\ge7$ without symmetry, a first nonradial rigidity classification for sequential multi-bubble dynamics. By constructing modified multi-bubble profiles and deploying a robust modulation framework, the authors reduce the infinite-dimensional PDE dynamics to finite-dimensional bubble parameters, then analyze three regimes determined by scale interactions and degeneracy: a dominant-bubble regime with a universal $t^{-2/(N-6)}$ speed for non-dominant bubbles, a totally non-degenerate regime with universal $t^{-1/(N-4)}$ scaling for all bubbles, and a minimally degenerate regime yielding a novel $t^{-1/(N-3)}$ speed. The results include a complete classification for up to four bubbles, with explicit constructions of minimally degenerate examples; the analysis hinges on coercivity of the linearized operators, precise inter-bubble interaction estimates via the matrix $A^*$, and a careful finite-dimensional reduction that tracks scales, centers, and signs. These findings extend previous radial results to the nonradial setting and reveal new blow-up rates arising from nontrivial bubble interactions, providing a framework that could extend to other critical parabolic systems.
Abstract
This paper concerns the classification of asymptotic behaviors in multi-bubble dynamics for the energy-critical nonlinear heat equations in large dimensions $N\geq7$ without symmetry. This multi-bubble dynamics appears naturally at least for a sequence of times in view of soliton resolution. We assume each bubble is given by the scalings and translations of $\pm W$ with (localized) non-colliding conditions for a sequence of times, where $W$ is the ground state. The case of one soliton was previously established and in particular there is no blow-up. We consider the case of $J\geq2$ solitons, where we expect only infinite-time blow-up. We are able to identify three different scenarios, where we have a continuous-in-time resolution with an unexpected universal blow-up speed. The first one is when one scaling is much larger than the others. In this case, one bubble does not concentrate (hence stabilize) and the other bubbles concentrate with the universal blow-up speed $t^{-2/(N-6)}$ together with strong sign constraints. Next, assuming we are not in the first scenario, we establish a non-degenerate condition on the positions of bubbles to obtain that all bubbles concentrate with the universal blow-up speed $t^{-1/(N-4)}$. The last case we consider is a degenerate, but not too much degenerate, scenario. Here again, we obtain that all bubbles concentrate with the universal blow-up speed $t^{-1/(N-3)}$. This last rate has not been discovered before. Our theorem covers the case of four or less bubbles and we provide the construction of examples. To our knowledge, this is the first classification result in the non-radial multi-bubble dynamics, where both the scales, positions, and signs enter the dynamics nontrivially.