{
  "id": "1507.07192",
  "version": "v1",
  "published": "2015-07-26T12:11:03.000Z",
  "updated": "2015-07-26T12:11:03.000Z",
  "title": "A Liouville-type theorem for the $3$-dimensional parabolic Gross-Pitaevskii and related systems",
  "authors": [
    "Quoc Hung Phan",
    "Philippe Souplet"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove a Liouville-type theorem for semilinear parabolic systems of the form $${\\partial_t u_i}-\\Delta u_i =\\sum_{j=1}^{m}\\beta_{ij} u_i^ru_j^{r+1}, \\quad i=1,2,...,m$$ in the whole space ${\\mathbb R}^N\\times {\\mathbb R}$. Very recently, Quittner [{\\em Math. Ann.}, DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for $m=2$ in dimension $N\\leq 2$, and partial results in higher dimensions in the range $p< N/(N-2)$. By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-V\\'eron, we partially improve the results of Quittner in dimensions $N\\geq 3$. In particular, our results solve the important case of the parabolic Gross-Pitaevskii system -- i.e. the cubic case $r=1$ -- in space dimension $N=3$, for any symmetric $(m,m)$-matrix $(\\beta_{ij})$ with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space ${\\mathbb R}^N_+\\times {\\mathbb R}$. As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-26T12:11:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville-type theorem",
      "dimensional parabolic gross-pitaevskii",
      "related systems",
      "corresponding initial value problem",
      "blow-up rate estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150707192P"
    }
  }
}