2020/05/13 by João da Silva, Pablo Ochoa, da Silva, Joao +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2005.06451
openalex publication_date 2020/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type (2 ≤ p< ∞) under a strong absorption condition: Δp u - (∂ u)/(∂ t) = λ0 u+q in ΩT \defeq Ω× (0, T), where 0 ≤ q < 1 and λ0 is a function bounded away from zero and infinity. 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 \mathfrakF0(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 provided their growth at infinity can be appropriately controlled. A specific analysis for Blow-up type solutions will be done as well. The results obtained in this article via our approach are new even for dead-core problems driven by the heat operator.