{
  "id": "math/0608290",
  "version": "v1",
  "published": "2006-08-11T17:50:38.000Z",
  "updated": "2006-08-11T17:50:38.000Z",
  "title": "Nonlinear evolution PDEs in R^+ \\times C^d: existence and uniqueness of solutions, asymptotic and Borel summability",
  "authors": [
    "O. Costin",
    "S. Tanveer"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a system of $n$-th order nonlinear quasilinear partial differential equations of the form $${\\bf u}_t + \\mathcal{P}(\\partial_{\\bf x}^{\\bf j}){\\bf u}+{\\bf g} \\left( {\\bf x}, t, \\{\\partial_{\\bf x}^{{\\bf j}} {\\bf u}\\}) =0; {\\bf {u}}({\\bf x}, 0) = {\\bf {u}}_I({\\bf x})$$ with $\\mathbf{u}\\in\\CC^{r}$, for $ t\\in (0,T)$ and large $|{\\bf x}|$ in a poly-sector $S$ in $\\mathbb{C}^d$ ($\\partial_{\\bf x}^{\\bf j} \\equiv \\partial_{x_1}^{j_1} \\partial_{x_2}^{j_2} ...\\partial_{x_d}^{j_d}$ and $j_1+...+j_d\\le n$). The principal part of the constant coefficient $n$-th order differential operator $\\mathcal{P}$ is subject to a cone condition. The nonlinearity ${\\bf g}$ and the functions $\\mb u_I$ and $\\mb u$ satisfy analyticity and decay assumptions in $S$.The paper shows existence and uniqueness of the solution of this problem and finds its asymptotic behavior for large $|\\bf x|$. Under further regularity conditions on $\\mb g$ and $\\mb u_I$ which ensure the existence of a formal asymptotic series solution for large $|\\mb x|$ to the problem, we prove its Borel summability (and automatically its asymptoticity) to an actual solution $\\mb u$.In special cases motivated by applications we show how the method can be adapted to obtain short-time existence, uniqueness and asymptotic behavior for small $t$,without size restriction on the space variable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-11T17:50:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35G25"
    ],
    "keywords": [
      "nonlinear evolution pdes",
      "borel summability",
      "asymptotic",
      "uniqueness",
      "nonlinear quasilinear partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007AnIHP..24..795C"
    }
  }
}