Monotone continuous dependence of solutions of singular quenching parabolic problems

Abstract

AbstractWe prove the continuous dependence, with respect to the initial datum of solutions of the “quenching parabolic problem”$$\partial _{t}u-\Delta u+\chi _{\{u>0\}}u^{-\beta }=\lambda u^{p}$$∂tu-Δu+χ{u>0}u-β=λup, with zero Dirichlet boundary conditions, when$$\beta \in (0,1),p\in (0,1],\lambda \ge 0$$β∈(0,1),p∈(0,1],λ≥0and$$\chi _{\{u>0\}}$$χ{u>0}denotes the characteristic function of the set of points (x, t) where$$u(x,t)>0$$u(x,t)>0. Notice that the absorption term$$\chi _{\{u>0\}}u^{-\beta }$$χ{u>0}u-βis singular and monotone decreasing which does not allow the application of standard monotonicity arguments.