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

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
JournalJournal of Differential Equations
DOIs
Publication statusAccepted/In press - 2020 Jan 1

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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