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On the weak Harnack inequality for unbounded non-negative super-solutions of degenerate double-phase parabolic equations

Mariia Savchenko, Igor Skrypnik, Yevgeniia Yevgenieva

Abstract

In the case $q> p\dfrac{n+2}{n}$, we give a proof of the weak Harnack inequality for non-negative super-solutions of degenerate double-phase parabolic equations under the additional assumption that $u\in L^{s}_{loc}(Ω_{T})$ with some $s >p\dfrac{n+2}{n}$.

On the weak Harnack inequality for unbounded non-negative super-solutions of degenerate double-phase parabolic equations

Abstract

In the case , we give a proof of the weak Harnack inequality for non-negative super-solutions of degenerate double-phase parabolic equations under the additional assumption that with some .
Paper Structure (14 sections, 15 theorems, 164 equations)

This paper contains 14 sections, 15 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Auxiliary material and integral estimates
  3. Local clustering Lemma
  4. De Giorgi--Poincare lemma
  5. Local energy estimates
  6. De Giorgi type lemma
  7. Expansion of positivity
  8. Proof of Proposition \ref{['pr3.1']} in the case ($i$)
  9. Proof of Proposition \ref{['pr3.1']} in the case ($ii$)
  10. Proof of Proposition \ref{['pr3.1']} under condition \ref{['eq3.8']}
  11. Expansion of positivity
  12. Weak Harnack inequality, proof of Theorem \ref{['th1.1']}
  13. Proof of Theorem \ref{['th1.1']} under the "hot" alternative case, condition \ref{['eq4.1']}
  14. Proof of Theorem \ref{['th1.1']} under the "cold" alternative case, condition \ref{['eq4.2']}

Key Result

Theorem 1.1

Let $u$ be a weak super-solution to equation eq1.1, let conditions eq1.2 and ($A$) be fulfilled. Assume additionally that $u \in L^{s}(\Omega_{T})$ and Then there exist positive constants $C_{1}$, $C_{2}$, $C_{3} >0$ depending only on $n$, $p$, $q$, $K_{1}$, $K_{2}$, $A$ and $d:=(\iint\limits_{\Omega_{T}} u^{s} dx dt)^{\frac{1}{s}}$ such that for a.a. $(x_{0}, t_{0}) \in \Omega_{T}$, either or

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Proposition 3.1
  • ...and 16 more