Decay of the Solutions of Parabolic Equations with Double Nonlinearity and the Degenerate Absorption Potential

Abstract

We study the behavior of solutions for the parabolic equation of nonstationary diffusion with double nonlinearity and a degenerate absorption term: $$ {\left({\left| u\right|}^{q-1} u\right)}_t-{\displaystyle \sum_{i=1}^N\frac{\partial }{\partial {x}_i}\left({\left|{\nabla}_x u\right|}^{q-1}\frac{\partial u}{\partial {x}_i}\right)+{a}_0(x){\left| u\right|}^{\lambda -1} u=0,} $$ where \( {a}_0(x)\ge {d}_0\; \exp \left(-\frac{\omega \left(\left| x\right|\right)}{{\left| x\right|}^{q+1}}\right) \) , d ₀ = const > 0, 0 ≤ λ < q, ω(⋅) ϵ C([0, + ∞)), ω(0) = 0, ω(τ) > 0 for τ > 0, and \( {\displaystyle {\int}_{0+}\frac{\omega \left(\tau \right)}{\tau} d\tau <\infty } \) . By the local energy method, we show that a Dini-type condition imposed on the function ω(·) guarantees the decay of an arbitrary solution for a finite period of time.