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

Abstract

We consider the high-dimensional equation ∂t⁡u-Δ⁢um+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];Lδ1⁢(Ω)){u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))}, with u-β⁢χ{u>0}∈L1⁢((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.

Keywords

• singular absorption and nonlinear diffusion equations
• pointwise gradient estimates
• quenching phenomenon
• free boundary
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