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 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)
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A semilinear parabolic equation with free boundary. / Choe, Hi Jun; Weiss, Georg Sebastian.
In: Indiana University Mathematics Journal, Vol. 52, No. 1, 01.01.2003, p. 19-50.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].
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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 -