Regularity for degenerate evolution equations with strong absorption

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=λ0u+qinΩT:=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=pp−1−q≥1+1p−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.