English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this manuscript, we study geometric regularity estimates for degenerateparabolic equations of $p$-Laplacian type ($2 \leq p< \infty$) under a strongabsorption 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 modelis interesting because it yields the formation of dead-core sets, i.e, regionswhere non-negative solutions vanish identically. We shall prove sharp andimproved parabolic $C^{\alpha}$ regularity estimates along the set$\mathfrak{F}_0(u, \Omega_T) = \partial \{u>0\} \cap \Omega_T$ (the freeboundary), where $\alpha= \frac{p}{p-1-q}\geq 1+\frac{1}{p-1}$. Some weakgeometric and measure theoretical properties as non-degeneracy, positivedensity, porosity and finite speed of propagation are proved. As anapplication, we prove a Liouville-type result for entire solutions providedtheir growth at infinity can be appropriately controlled. A specific analysisfor Blow-up type solutions will be done as well. The results obtained in thisarticle via our approach are new even for dead-core problems driven by the heatoperator.

Regularity for degenerate evolution equations with strong absorption