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Local Continuity and Harnack’s Inequality for Double-Phase Parabolic Equations

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Abstract

We consider parabolic equations of the form

$$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$

In the range \(\frac {2n}{n+1}<p< q\) we establish local properties of bounded solutions u to equation as local continuity and Harnack’s type inequality. These properties in a neighborhood of a point (x0, t0) depend on the value of the function a(x0, t0).

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Acknowledgments

This work is supported by grants of Ministry of Education and Science of Ukraine, project numbers are 0118U003138, 0119U100421. We thank the referee for the careful reading of the preliminary version of the paper and suggestions for an improved presentation.

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Authors and Affiliations

  1. Vasyl’ Stus Donetsk National University, 600-richchia Str. 21, 21021, Vinnytsia, Ukraine

    Kateryna O. Buryachenko & Igor I. Skrypnik

  2. Institute of Applied Mathematics and Mechanics of NAS of Ukraine, G. Batiouk str. 19, 84100, Sloviansk, Ukraine

    Igor I. Skrypnik

Authors
  1. Kateryna O. Buryachenko
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  2. Igor I. Skrypnik
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Correspondence to Igor I. Skrypnik.

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Buryachenko, K.O., Skrypnik, I.I. Local Continuity and Harnack’s Inequality for Double-Phase Parabolic Equations. Potential Anal 56, 137–164 (2022). https://doi.org/10.1007/s11118-020-09879-9

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  • DOI: https://doi.org/10.1007/s11118-020-09879-9

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