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 ($2 \leq p< \infty$) under a strong absorption condition: $ \Delta_p u - \frac{\partial u}{\partial t} = \lambda_0 u_{+}^q \quad \mbox{in} \quad \Omega_T \defeq \Omega \times (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 $\alpha= \frac{p}{p-1-q}\geq 1+\frac{1}{p-1}$. 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.