{
  "id": "2305.19364",
  "version": "v1",
  "published": "2023-05-30T19:12:01.000Z",
  "updated": "2023-05-30T19:12:01.000Z",
  "title": "Existence of solutions for a $k$-Hessian equation and its connection with self-similar solutions",
  "authors": [
    "Justino Sánchez"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\alpha,\\beta$ be real parameters and let $a>0$. We study radially symmetric solutions of \\begin{equation*} S_k(D^2v)+\\alpha v+\\beta \\xi\\cdot\\nabla v=0,\\, v>0\\;\\; \\mbox{in}\\;\\; \\mathbb{R}^n,\\; v(0)=a, \\end{equation*} where $S_k(D^2v)$ denotes the $k$-Hessian operator of $v$. For $\\alpha\\leq\\frac{\\beta(n-2k)}{k}\\;\\;\\mbox{and}\\;\\;\\beta>0$, we prove the existence of a unique solution to this problem, without using the phase plane method. We also prove existence and properties of the solutions of the above equation for other ranges of the parameters $\\alpha$ and $\\beta$. These results are then applied to construct different types of explicit solutions, in self-similar forms, to a related evolution equation. In particular, for the heat equation, we have found a new family of self-similar solutions of type II which blows up in finite time. These solutions are represented as a power series, called the Kummer function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-30T19:12:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar solutions",
      "hessian equation",
      "connection",
      "study radially symmetric solutions",
      "phase plane method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}