Regularity for degenerate evolution equations with strong absorption

Abstract

In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of $p$-Laplacian type ($2\le p<\infty$) under a strong absorption condition: $ Δ_p u - \frac{\partial u}{\partial t} = λ_0 u_{+}^q \quad \mbox{in} \quad Ω_T \defeq Ω\times (0, T), $ where $0\le q<1$ 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 $\alpha =\frac{p}{p-1-q}\ge 1+\frac{1}{p-1}$. 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. A 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.

References in corpus (1)

• Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries