Regularity for degenerate evolution equations with strong absortion

JOAO VITOR DA SILVA; PABLO OCHOA; ANALÍA 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 an 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 $0 leq q < 1$ and $lambda_0$ 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 $C^{alpha}$ 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 provided their growth at infinity can be appropriately controlled. An 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