In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type (2 p < •) under a strong absorption condition:This model is interesting because it yields the formation of dead-core sets, i.e, regions where nonnegative solutions vanish identically. We shall prove sharp and improved parabolic C a regularity estimates along the set F 0 (u, W T ) = ∂ {u > 0} \ W T (the free boundary), where a = 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.2000 Mathematics Subject Classification. 35B53, 35B65, 35J60, 35K55, 35K65. Key words and phrases. p-Laplacian type operators, dead-core problems, sharp and improved intrinsic regularity, Liouville type results.
In this manuscript, we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower order term depending on x, u and ∇u. More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending results presented in Juutinen, Lindqvist and Manfredi [9], and Julin and Juutinen [6]. Moreover, we provide a converse statement in the full case under extra assumptions on the data.
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