We consider the high-dimensional equation {\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\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))}, with {u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)}, {\delta(x)=d(x,\partial\Omega)}, we prove some pointwise gradient estimates for a certain range of the dimension N, {m\geq 1} and {\beta\in(0,m)}, mainly when the absorption dominates over the diffusion ({1\leq m<2+\beta}).
In particular, a new kind of universal gradient estimate is proved when {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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.