### Abstract

We consider non-negative solutions of the heat equation with strong absorption, ∂_{t}u - Δu = -u^{gamma;}χ_{{u>0}} in (0, ∞) × Ω, where Ω is a smooth bounded domain in ℝ^{n}, γ ∈ [0, 1), and initial and boundary data are prescribed. Assuming merely regularity and a growth condition of the data, we prove optimal regularity and non-degeneracy estimates for the solution, which already have interesting consequences as for example finite propagation speed of the set {u > 0}. We then show that the n + 1-dimensional Hausdorff measure with respect to the parabolic metric is locally finite on the free boundary ∂ {u > 0}. Concerning the Cauchy problem with respect to γ ∈ (0, 1) we know more: any self-similar solution in (- ∞, 0) × ℝ^{n} is either time-independent or coincides with the solution U_{1}(t, x) = max(0,(1 - γ)(-t))^{1/(1-γ)}. As a consequence, the free boundary can be divided into a closed set of horizontal points which is locally contained in an n-dimensional Lipschitz surface and on which U_{1} is the unique blow-up limit, and a relatively open set of non-horizontal points on which any blow-up limit is a steady-state solution. We proceed to characterize the asymptotic behavior near horizontal points. Finally we consider the case of one space dimension in which we obtain that any blow-up limit is unique and that the regular non-horizontal part of the free boundary is open and a C^{1/2} -surface. The last result is extended to the case of higher dimensions in [16].

Original language | English |
---|---|

Pages (from-to) | 19-50 |

Number of pages | 32 |

Journal | Indiana University Mathematics Journal |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2003 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*52*(1), 19-50. https://doi.org/10.1512/iumj.2003.52.2124

}

*Indiana University Mathematics Journal*, vol. 52, no. 1, pp. 19-50. https://doi.org/10.1512/iumj.2003.52.2124

**A semilinear parabolic equation with free boundary.** / Choe, Hi Jun; Weiss, Georg Sebastian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A semilinear parabolic equation with free boundary

AU - Choe, Hi Jun

AU - Weiss, Georg Sebastian

PY - 2003/1/1

Y1 - 2003/1/1

N2 - We consider non-negative solutions of the heat equation with strong absorption, ∂tu - Δu = -ugamma;χ{u>0} in (0, ∞) × Ω, where Ω is a smooth bounded domain in ℝn, γ ∈ [0, 1), and initial and boundary data are prescribed. Assuming merely regularity and a growth condition of the data, we prove optimal regularity and non-degeneracy estimates for the solution, which already have interesting consequences as for example finite propagation speed of the set {u > 0}. We then show that the n + 1-dimensional Hausdorff measure with respect to the parabolic metric is locally finite on the free boundary ∂ {u > 0}. Concerning the Cauchy problem with respect to γ ∈ (0, 1) we know more: any self-similar solution in (- ∞, 0) × ℝn is either time-independent or coincides with the solution U1(t, x) = max(0,(1 - γ)(-t))1/(1-γ). As a consequence, the free boundary can be divided into a closed set of horizontal points which is locally contained in an n-dimensional Lipschitz surface and on which U1 is the unique blow-up limit, and a relatively open set of non-horizontal points on which any blow-up limit is a steady-state solution. We proceed to characterize the asymptotic behavior near horizontal points. Finally we consider the case of one space dimension in which we obtain that any blow-up limit is unique and that the regular non-horizontal part of the free boundary is open and a C1/2 -surface. The last result is extended to the case of higher dimensions in [16].

AB - We consider non-negative solutions of the heat equation with strong absorption, ∂tu - Δu = -ugamma;χ{u>0} in (0, ∞) × Ω, where Ω is a smooth bounded domain in ℝn, γ ∈ [0, 1), and initial and boundary data are prescribed. Assuming merely regularity and a growth condition of the data, we prove optimal regularity and non-degeneracy estimates for the solution, which already have interesting consequences as for example finite propagation speed of the set {u > 0}. We then show that the n + 1-dimensional Hausdorff measure with respect to the parabolic metric is locally finite on the free boundary ∂ {u > 0}. Concerning the Cauchy problem with respect to γ ∈ (0, 1) we know more: any self-similar solution in (- ∞, 0) × ℝn is either time-independent or coincides with the solution U1(t, x) = max(0,(1 - γ)(-t))1/(1-γ). As a consequence, the free boundary can be divided into a closed set of horizontal points which is locally contained in an n-dimensional Lipschitz surface and on which U1 is the unique blow-up limit, and a relatively open set of non-horizontal points on which any blow-up limit is a steady-state solution. We proceed to characterize the asymptotic behavior near horizontal points. Finally we consider the case of one space dimension in which we obtain that any blow-up limit is unique and that the regular non-horizontal part of the free boundary is open and a C1/2 -surface. The last result is extended to the case of higher dimensions in [16].

UR - http://www.scopus.com/inward/record.url?scp=0038400195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038400195&partnerID=8YFLogxK

U2 - 10.1512/iumj.2003.52.2124

DO - 10.1512/iumj.2003.52.2124

M3 - Article

VL - 52

SP - 19

EP - 50

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 1

ER -