Regularity for degenerate evolution equations with strong absorption

ANALÍA SILVA; PABLO OCHOA; JOAO VITOR DA SILVA

JOURNAL OF DIFFERENTIAL EQUATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2018 vol. 264 p. 7270 - 7293

0022-0396

In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type ($2 leq p< infty$) under a strong absorption condition: $$Delta_p u - rac{partial u}{partial t} = lambda_0 u_{+}^q quad mbox{in} quad Omega_T defeq Omega imes (0, T),$$where $0 leq q < 1$. 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 $C^{alpha}$ regularity estimates along the set $mathfrak{F}_0(u, Omega_T) = partial {u>0} cap Omega_T$ (the free boundary), where $alpha= rac{p}{p-1-q}geq 1+rac{1}{p-1}$. 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.