{
  "id": "2311.07154",
  "version": "v1",
  "published": "2023-11-13T08:50:34.000Z",
  "updated": "2023-11-13T08:50:34.000Z",
  "title": "On the instability of threshold solutions of reaction-diffusion equations, and applications to optimization problems",
  "authors": [
    "Grégoire Nadin"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\\Delta u = f(x,u)$ on $(0,\\infty)\\times \\mathbb{R}^N$, where the initial datum $u(0,x)=u_0(x)$ is such that $\\lim_{t\\to +\\infty} u(t,x)=W(x)$ for all $x$, with $W$ a steady state in $H^1(\\mathbb{R}^N)$. We characterize the perturbations $h$ such that, if $u^h$ is the solution associated with the initial datum $u_0+h$, then, if $h$ is small enough in a sense, one has $u^h(t,x)>W(x)$ (resp. $u(t,x)<W(x)$) for $t$ large. This condition depends on the sign of $\\int_{\\mathbb{R}^N} h(x)p(0,x)dx$, where $p$ is an adjoint solution, which satisfies a backward parabolic equation on $(0,\\infty)$ and is uniquely defined [7]. We then provide two applications of our result. We first address an open problem stated in [8] when $N=1$ and $f$ is a bistable nonlinearity independent of $x$. Namely, we compute the derivative of the critical length $L^*(r)$ associated with the initial datum $\\mathbf{I}_{(-L-r,-r)\\cup (r,L+r)}$, that is the length $L$ above (resp. below) which $u(t,x)$ converges to $1$ (resp. $0$) as $t\\to +\\infty$. Lastly, again when $N=1$ and $f$ is a bistable nonlinearity independent of $x$, we prove the existence and characterize with a bathtub principle the initial datum $\\underline{u}_0$ minimizing some cost function $\\int_\\mathbb{R} j(u_0)$ and guaranteeing at the same time that $\\liminf_{t\\to +\\infty} u(t,x)>0$ for all $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-13T08:50:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reaction-diffusion equation",
      "optimization problems",
      "initial datum",
      "threshold solutions",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}