We study the behavior of solutions for the parabolic equation of nonstationary diffusion with double nonlinearity and a degenerate absorption term:
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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 1, pp. 89–107, January, 2014.
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Stepanova, E.V. Decay of the Solutions of Parabolic Equations with Double Nonlinearity and the Degenerate Absorption Potential. Ukr Math J 66, 99–121 (2014). https://doi.org/10.1007/s11253-014-0915-x
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DOI: https://doi.org/10.1007/s11253-014-0915-x