Abstract We consider the high-dimensional equation ∂ t ⁡ u - Δ ⁢ u m + u - β ⁢ χ { u > 0 } = 0 {\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0} , extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution u ∈ 𝒞 ⁢ ( [ 0 , T ] ; L δ 1 ⁢ ( Ω ) ) {u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))} , with u - β ⁢ χ { u > 0 } ∈ L 1 ⁢ ( ( 0 , T ) × Ω ) {u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)} , δ ⁢ ( x ) = d ⁢ ( x , ∂ ⁡ Ω ) {\delta(x)=d(x,\partial\Omega)} , we prove some pointwise gradient estimates for a certain range of the dimension N, m ≥ 1 {m\geq 1} and β ∈ ( 0 , m ) {\beta\in(0,m)} , mainly when the absorption dominates over the diffusion ( 1 ≤ m < 2 + β {1\leq m<2+\beta} ). In particular, a new kind of universal gradient estimate is proved when m + β ≤ 2 {m+\beta\leq 2} . Several qualitative properties (such as the finite time quenching phenomena and the finite speed of propagation) and the study of the Cauchy problem are also considered.

中文翻译：

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具有奇异淬火项的多维慢扩散方程的逐点梯度估计

摘要 我们考虑高维方程∂ t ⁡ u - Δ ⁢ um + u - β ⁢ χ { u > 0 } = 0 {\partial_{t}u-\Delta u^{m}+u^{-\ beta}{\chi_{\{u>0\}}}=0} ，扩展了 B. Kawohl 和 R. Kersner 在 1992 年对一维情况进行的数学处理。除了存在一个非常弱的解 u ∈ 𝒞 ⁢ ( [ 0 , T ] ; L δ 1 ⁢ ( Ω ) ) {u\in\mathcal{C}([0,T];L_{\delta}^{ 1}(\Omega))} , u - β ⁢ χ { u > 0 } ∈ L 1 ⁢ ( ( 0 , T ) × Ω ) {u^{-\beta}\chi_{\{u>0\ }}\in L^{1}((0,T)\times\Omega)} , δ ⁢ ( x ) = d ⁢ ( x , ∂ ⁡ Ω ) {\delta(x)=d(x,\partial \Omega)} ，我们证明了一定范围的维度 N 的逐点梯度估计，m ≥ 1 {m\geq 1} 和 β ∈ ( 0 , m ) {\beta\in(0,m)} ，主要是当吸收支配扩散时（ 1 ≤ m < 2 + β {1\leq m<2+\beta} ）。特别是，当 m + β ≤ 2 {m+\beta\leq 2} 时，证明了一种新的通用梯度估计。还考虑了几个定性属性（例如有限时间猝灭现象和有限传播速度）和柯西问题的研究。