{
  "id": "2404.09410",
  "version": "v1",
  "published": "2024-04-15T01:55:25.000Z",
  "updated": "2024-04-15T01:55:25.000Z",
  "title": "$L^2$-based stability of blowup with log correction for semilinear heat equation",
  "authors": [
    "Thomas Y. Hou",
    "Van Tien Nguyen",
    "Yixuan Wang"
  ],
  "comment": "22 pages, 4 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We propose an alternative proof of the classical result of type-I blowup with log correction for the semilinear equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbation around the approximate profile and establish a weighted $H^k$ stability, thereby avoiding the use of a topological argument and the analysis of a linearized spectrum. Therefore, this approach can be adopted even if we only have a numerical profile and do not have explicit information on the spectrum of its linearized operator. This result generalizes the $L^2$-based stability argument to blowups that are not exactly self-similar and can be adapted to higher dimensions. Numerical results corroborate the effectiveness of our normalization, even in the large perturbation regime beyond our theoretical setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-15T01:55:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semilinear heat equation",
      "log correction",
      "large perturbation regime",
      "semilinear equation",
      "explicit information"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}