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Journal of Mathematical Sciences

, Volume 233, Issue 6, pp 807–827 | Cite as

On the Stabilization Rate of Solutions of the Cauchy Problem for a Parabolic Equation with Lower-Order Terms

  • V. N. Denisov
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Abstract

The following Cauchy problem for parabolic equations is considered in the half-space \( \overline{D}={\mathrm{\mathbb{R}}}^N\times \left[0,\infty \right) \), N ≥ 3:
$$ {L}_1u\equiv Lu+c\left(x,t\right)u-{u}_t=0,\kern0.5em \left(x,t\right)\in D,\kern0.5em u\left(x,0\right)={u}_0(x),\kern0.5em x\in {\mathrm{\mathbb{R}}}^N. $$

It is proved that for any bounded and continuous in ℝN initial function u0(x), the solution of the above Cauchy problem stabilizes to zero uniformly with respect to x from any compact set K in ℝN either exponentially or as a power (depending on the estimate for the coefficient c(x, t) of the equation).

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

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