Weighted Lq(Lp)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

Beom Seok Han, Kyeong Hun Kim, Daehan Park

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We present a weighted Lq(Lp)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation ∂t αu(t,x)=Δu(t,x)+f(t,x),t>0,x∈Rd. Here, α∈(0,2) and ∂t α is the Caputo fractional derivative of order α. In particular we prove that for any p,q∈(1,∞), w1(x)∈Ap and w2(t)∈Aq, ∫0∞(∫Rd|uxx|pw1dx)q/pw2dt≤N∫0∞(∫Rd|f|pw1dx)q/pw2dt, where Ap is the class of Muckenhoupt Ap weights. Our approach is based on the sharp function estimates of the derivatives of solutions.

Original languageEnglish
Pages (from-to)3515-3550
Number of pages36
JournalJournal of Differential Equations
Volume269
Issue number4
DOIs
Publication statusPublished - 2020 Aug 5

Keywords

  • Caputo fractional derivative
  • Fractional diffusion-wave equation
  • L(L)-theory
  • Muckenhoupt A weights

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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