{
  "abstract": "We study the local existence and blow-up behavior of solutions, with possible growth at space infinity, for the diffusive Hamilton-Jacobi equation $u_t-\\Delta u=|\\nabla u|^p$ ($p>1$), in a half-space with Dirichlet boundary conditions. Under optimal, polynomial growth assumptions on the initial data, characterized by the critical exponent $p/(p-1)$, we first prove local existence and uniqueness of a maximal classical solution and establish a blow-up alternative. This is complemented by a nonexistence result showing that $p/(p-1)$ is the sharp threshold for admissible growth at infinity. Next, for initial data with subcritical growth, we show global-in-time existence for $1<p\\le2$ and that finite-time blow-up can occur only through gradient blow-up for $p>2$, and we derive sharp upper and lower estimates on the gradient, including a precise type~II boundary blow-up rate and a dichotomy in its behavior near the singular time. Finally, in the case $p>2$, for any type~II blow-up solution, we prove convergence (after rescaling) to an explicit one-dimensional profile and providing a refined description of the asymptotic singularity formation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.25624",
  "author_total": 1,
  "authors": [
    {
      "name": "Loth Damagui Chabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ap4br4ezjbv1n.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "ap4br4ezjbv1n",
  "issns": [],
  "landing_url": "https://hal.science/hal-05801886v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05801886/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ap4br4ezjbv1n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:58Z",
      "native_id": "hal-05801886",
      "source": "hal",
      "url": "https://hal.science/hal-05801886v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data",
  "title_text": "Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:58Z",
  "url": "https://research.draconitic.com/paper/ap4br4ezjbv1n",
  "venue": "HAL",
  "versions": []
}