A semilinear parabolic equation with free boundary

We consider non-negative solutions of the heat equation with strong absorption, ∂_tu - Δu = -u^gamma;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 C^1/2 -surface. The last result is extended to the case of higher dimensions in [16].