论文标题
具有强吸收的退化进化方程的规律性
Regularity for degenerate evolution equations with strong absorption
论文作者
论文摘要
在本手稿中，我们研究了$ p $ -laplacian类型的退化抛物线方程（$ 2 \ leq p <\ infty $）的几何规律性估计值：$ Δ_pu- \ frac {\ partial u} {\ partial t} =λ_0u _ {+}^q \ quad \ mbox {in} \ quadω_t\ quadω_t\ defeqω\ times（0，t），$ 其中$ 0 \ leq q <1 $和$λ_0$是一个函数，远离零和无穷大。该模型很有趣，因为它产生了死核集的形成，即非负解决方案相同消失的区域。我们将在集合$ \ mathfrak {f} _0（u，ω_t）= \ partial \ {u> 0 \ {u> 0 \} \capΩ_t$（free Boundard）中，沿$ \ mathfrak {f} _0（u，ω_t）=敏锐而改进的抛物面$ c^α$规则性估算1+ \ frac {1} {p-1} $。证明了某些弱的几何形状和理论特性作为非分类，正密度，孔隙率和有限的传播速度。作为应用程序，我们证明了整个解决方案的liouville型结果，只要它们在无穷大的生长就可以得到适当控制。对爆破型解决方案的特定分析也将进行。即使是由于热算子驱动的死核问题，本文在本文中获得的结果也是新的。
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: $ Δ_p u - \frac{\partial u}{\partial t} = λ_0 u_{+}^q \quad \mbox{in} \quad Ω_T \defeq Ω\times (0, T), $ where $0 \leq q < 1$ and $λ_0$ 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 $C^α$ regularity estimates along the set $\mathfrak{F}_0(u, Ω_T) = \partial \{u>0\} \cap Ω_T$ (the free boundary), where $α= \frac{p}{p-1-q}\geq 1+\frac{1}{p-1}$. 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.
