Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term

Abstract

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.