Stable Type-I Blowup by Local Normalization Conditions: Nonlinear Heat and Complex Ginzburg-Landau Equations
Time: 2024-09-09
Published By: Wenqiong Li
Speaker(s): Yixuan Wang (California Institute of Technology)
Time: 16:00-17:00 September 19, 2024
Venue: Room 78301, Jingchunyuan 78, BICMR
We provide a general framework that could be potentially amenable to the stability of a large class of problems with type I singularities in a weighted Sobolev space and use nonlinear heat and complex Ginzburg-Landau equations as examples, which are joint works with Tom Hou, Van Tien Nguyen, and Jiajie Chen. A generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel additional modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose vanishing conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. We then establish the full stability of the blowup using weighted energy estimates, without relying on a topological argument or a spectrum analysis. The log correction for the blowup rate is automatically inferred by refined estimates of the modulation parameters.
