{
  "id": "1906.06625",
  "version": "v1",
  "published": "2019-06-15T22:46:40.000Z",
  "updated": "2019-06-15T22:46:40.000Z",
  "title": "Some results for the large time behavior of Hamilton-Jacobi Equations with Caputo Time Derivative",
  "authors": [
    "Olivier Ley",
    "Erwin Topp",
    "Miguel Yangari"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We obtain some H\\\"older regularity estimates for an Hamilton-Jacobi with fractional time derivative of order $\\alpha \\in (0,1)$ cast by a Caputo derivative. The H\\\"older seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case $\\alpha=1$, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-15T22:46:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large time behavior",
      "caputo time derivative",
      "hamilton-jacobi equations",
      "regularity estimates",
      "establish partial results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}