Harnack's inequality for degenerate double phase parabolic equations under the non-logarithmic Zhikov's condition

We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with \((p,q)\) growth $$ u_{t}-{\rm div}\left(\mid \nabla u \mid^{p-2}\nabla u + a(x,t) \mid \nabla u \mid^{q-2}\nabla u \right)=0,\quad a(x,t) \geq 0 , $$ under the generalized non-logarit...

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Bibliographic Details
Published in:arXiv.org 2023-04
Main Authors: Savchenko, Mariia, Skrypnik, Igor, Yevgenieva, Yevgeniia
Format: Article
Language:English
Subjects:
Logarithms
Mathematical analysis
Online Access:Get full text
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Summary:We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with \((p,q)\) growth $$ u_{t}-{\rm div}\left(\mid \nabla u \mid^{p-2}\nabla u + a(x,t) \mid \nabla u \mid^{q-2}\nabla u \right)=0,\quad a(x,t) \geq 0 , $$ under the generalized non-logarithmic Zhikovs conditions $$ \mid a(x,t)-a(y,\tau)\mid \leqslant A\mu(r) r^{q-p},\quad (x,t),(y,\tau)\in Q_{r,r}(x_{0},t_{0}),$$ $$\lim\limits_{r\rightarrow 0}\mu(r) r^{q-p}=0,\quad \lim\limits_{r\rightarrow 0}\mu(r)=+\infty,\quad \int\limits_{0} \mu^{-\beta}(r)\frac{dr}{r} =+\infty,$$ with some \(\beta >0\).
ISSN:2331-8422