A semilinear parabolic equation with free boundary

Hi Jun Choe, Georg Sebastian Weiss

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7 Citations (Scopus)

Abstract

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].

Original languageEnglish
Pages (from-to)19-50
Number of pages32
JournalIndiana University Mathematics Journal
Volume52
Issue number1
DOIs
Publication statusPublished - 2003 Jan 1

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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