Regularity for degenerate evolution equations with strong absorption

Abstract

In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type (2 ≤ p < ∞) under a strong absorption condition: Δpu − ∂u/∂t = λ0uq + in ΩT := Ω × (0,T), where 0 ≤ q < 1. This model is interesting because it yields the formation of dead-core sets, i.e., regions where non-negative solutions vanish identically. We shall prove sharp and improved parabolic Cα regularity estimates along the set F0(u, ΩT) = ∂{u > 0} ∩ΩT (the free boundary), where α = p/p−1−q ≥ 1 + 1/p−1. Some weak geometric and measure theoretical properties as non-degeneracy, positive density, porosity and finite speed of propagation are proved. As an application, we prove a Liouville-type result for entire solutions. A specific analysis for Blow-up type solutions will be done as well. The results are new even for dead-core problems driven by the heat operator.