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≥vt-A(v)-|v|q-1v${ u_t - A(u)-|u|^{q-1}u \ge v_t - A (v)-|v|^{q-1}v }$ on the set 𝕊τ=(τ,+∞)×ℝn${{\mathbb {S}}_{\tau }= (\tau , +\infty )\times {\mathbb {R}}^n}$, where q > 0, n ≥ 1, τ is a real number or τ=-∞${\tau =-\infty }$, and the differential operator A satisfies the α-monotonicity condition. Model examples of the operator A in our study are the well-known p-Laplacian operator defined by the relation Δp(w)=divx(|∇xw|p-2∇xw)${\Delta _p (w)=\operatorname{div}_x(|\nabla _x w|^{p-2}\nabla _x w) }$ and its well-known modification defined by Δ˜p(w)=∑i=1n∂∂xi(|∂w∂xi|p-2∂w∂xi)${ \widetilde{\Delta }_p (w) = \sum _{i=1}^n\frac{\partial }{\partial x_i}( |\frac{\partial w}{\partial x_i}|^{p-2}\frac{\partial w}{\partial x_i})}$.

Keywords

• liouville theorem
• comparison principle
• weak solution
• quasilinear singular parabolic inequality
• 35k59
• 35b51
• 35b53
• 35b08
• 35r45
• 35k67