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ABSTRACT. We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the formwhere F is elliptic with respect to the Hessian argument and f ∈ L p,q (Q 1 ). The quantity κ(n, p,q) := n p + 2 q determines to which regularity regime a solution of (Eq) belongs. We prove that when 1 < κ(n, p,q) < 2 − ε F , solutions are parabolic-Hölder continuous for a sharp, quantitative exponent 0 < α(n, p,q) < 1. Precisely at the critical borderline case, κ(n, p,q) = 1, we obtain sharp Log-Lipschitz regularity estimates. When 0 < κ(n, p,q) < 1, solutions are locally of class C 1+σ, 1+σ 2 and in the limiting case κ(n, p,q) = 0, we show C 1,Log-Lip regularity estimates provided F has "better" a priori estimates.

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.

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