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

Beom Seok Han; Kyeong Hun Kim; Daehan Park

doi:10.1016/j.jde.2020.03.005

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

Beom Seok Han, Kyeong Hun Kim, Daehan Park

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

Abstract

We present a weighted L_q(L_p)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation ∂_t ^αu(t,x)=Δu(t,x)+f(t,x),t>0,x∈R^d. 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)∈A_p and w₂(t)∈A_q, ∫0∞(∫R^d|u_xx|^pw₁dx)^q/pw₂dt≤N∫0∞(∫R^d|f|^pw₁dx)^q/pw₂dt, where A_p is the class of Muckenhoupt A_p weights. Our approach is based on the sharp function estimates of the derivatives of solutions.

Original language	English
Pages (from-to)	3515-3550
Number of pages	36
Journal	Journal of Differential Equations
Volume	269
Issue number	4
DOIs	https://doi.org/10.1016/j.jde.2020.03.005
Publication status	Published - 2020 Aug 5

Bibliographical note

Funding Information:
The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324).

Publisher Copyright:
© 2020 Elsevier Inc.

Keywords

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

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jde.2020.03.005

Cite this

@article{abe4ff39516644b9b529393d8fc6466f,

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

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.",

keywords = "Caputo fractional derivative, Fractional diffusion-wave equation, L(L)-theory, Muckenhoupt A weights",

author = "Han, {Beom Seok} and Kim, {Kyeong Hun} and Daehan Park",

note = "Funding Information: The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324). Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2020",

month = aug,

day = "5",

doi = "10.1016/j.jde.2020.03.005",

language = "English",

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T1 - Weighted Lq(Lp)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

AU - Han, Beom Seok

AU - Kim, Kyeong Hun

AU - Park, Daehan

N1 - Funding Information: The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324). Publisher Copyright: © 2020 Elsevier Inc.

PY - 2020/8/5

Y1 - 2020/8/5

N2 - 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.

AB - 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.

KW - Caputo fractional derivative

KW - Fractional diffusion-wave equation

KW - L(L)-theory

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