Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term
Nguyen Anh Dao, Jesús Ildefonso Díaz Díaz, Quan Ba Hong Nguyen · Advanced Nonlinear Studies · 2020
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,\partia