Monotone continuous dependence of solutions of singular quenching parabolic problems

Abstract

We prove the continuous dependence, with respect to the initial datum of solutions of the “quenching parabolic problem” $$\partial _{t}u-\Delta u+\chi _{\{u>0\}}u^{-\beta }=\lambda u^{p}$$ ∂ t u - Δ u + χ { u > 0 } u - β = λ u p , with zero Dirichlet boundary conditions, when $$\beta \in (0,1),p\in (0,1],\lambda \ge 0$$ β ∈ ( 0 , 1 ) , p ∈ ( 0 , 1 ] , λ ≥ 0 and $$\chi _{\{u>0\}}$$ 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 - β is singular and monotone decreasing which does not allow the application of standard monotonicity arguments.

Singular anisotropic equations with a sign-changing perturbation

“…We prove a multiplicity theorem producing two nontrivial smooth solutions when θ ∞ is small. We should also mention the related isotropic (constant exponents) works of Arora [1], Haddaoui et al [8], Diaz and Giacomoni [4], Papageorgiou, Vetro, and Vetro [14] (balanced growth problems), Kumar, Rȃdulescu, and Sreenadh [10] (unbalanced growth problems) and the anisotropic work on systems of Leggat and Miri [11].…”