{
  "id": "2107.10442",
  "version": "v1",
  "published": "2021-07-22T03:40:44.000Z",
  "updated": "2021-07-22T03:40:44.000Z",
  "title": "The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces",
  "authors": [
    "Yingying Guo"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg-Whitham equation in both supercritical Besov spaces $B^s_{p,r},\\ s>1+\\frac{1}{p},\\ 1\\leq p,r\\leq+\\infty$ and critical Besov spaces $B^{1+\\frac{1}{p}}_{p,1},\\ 1\\leq p<+\\infty$, which improves the previous work \\cite{y2,ho,ht}. Then, we prove the solution is not uniformly continuous dependence on the initial data in supercritical Besov spaces $B^s_{p,r},\\ s>1+\\frac{1}{p},\\ 1\\leq p\\leq+\\infty,\\ 1\\leq r<+\\infty$ and critical Besov spaces $B^{1+\\frac{1}{p}}_{p,1},\\ 1\\leq p<+\\infty$. At last, we show that the solution is ill-posed in $B^{\\sigma}_{p,\\infty}$ with $\\sigma>3+\\frac{1}{p},\\ 1\\leq p\\leq+\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-22T03:40:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "initial data",
      "fornberg-whitham equation",
      "non-uniform dependence",
      "supercritical besov spaces",
      "ill-posedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}