2017
Том 69
№ 9

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

Stepanova E. V.

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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.

English version (Springer): Ukrainian Mathematical Journal 66 (2014), no. 1, pp 99-121.

Citation Example: Stepanova E. V. Decay of the Solutions of Parabolic Equations with Double Nonlinearity and the Degenerate Absorption Potential // Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 89–107.

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