{
  "id": "1501.04251",
  "version": "v1",
  "published": "2015-01-18T00:38:35.000Z",
  "updated": "2015-01-18T00:38:35.000Z",
  "title": "The one-dimensional heat equation in the Alexiewicz norm",
  "authors": [
    "Erik Talvila"
  ],
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock--Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let $\\Theta_t(x)=\\exp(-x^2/(4t))/\\sqrt{4\\pi t}$ be the heat kernel. With initial data $f$ that is the distributional derivative of a continuous function, it is shown that $u_t(x):=u(x,t):=f\\ast\\Theta_t(x)$ is a classical solution of the heat equation $u_{11}=u_2$. The estimate $\\|f\\ast\\Theta_t\\|_\\infty\\leq\\|f\\|/\\sqrt{\\pi t}$ holds. The Alexiewicz norm is $\\|f\\|=\\sup_I|\\int_If|$, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, $\\|u_t-f\\|\\to 0$ as $t\\to 0^+$. The solution of the heat equation is unique under the assumptions that $\\|u_t\\|$ is bounded and $u_t\\to f$ in the Alexiewicz norm for some integrable $f$. The heat equation is also considered with initial data that is the $n$th derivative of a continuous function and in weighted spaces such that $\\int_{-\\infty}^\\infty f(x)\\exp(-ax^2)\\,dx$ exists for some $a>0$. Similar results are obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-18T00:38:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K05",
      "46F05",
      "26A39",
      "46F10",
      "46G12"
    ],
    "keywords": [
      "one-dimensional heat equation",
      "alexiewicz norm",
      "initial data",
      "real line",
      "continuous primitive integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150104251T"
    }
  }
}