A Liouville comparison principle for solutions of quasilinear singular parabolic inequalities

Abstract

We obtain a Liouville comparison principle for entire weak solutions (u, V) of quasilinear singular parabolic second-order partial differential inequalities of the form ut - A(u) - |u|q-1u ≥νt- A(ν) - |u|q-1 ν on the set Sτ = (τ, +∞) × ℝn, where q > 0, n ≥ 1, τ is a real number or τ = - ∞, and the differential operator A satisfies the α-monotonicity condition. Model examples of the operator in our study are the well-known p-Laplacian operator defined by the relation δ(ω) = divx(|∇xω|p-2∇xω) and its well-known modification denned by δp (ω) = Σni=1ρ/ρxi(|ρω/ρxi|p-2ρω/ρxi).