A weighted sobolev space theory for the diffusion-wave equations with time-fractional derivatives on C1 domains

Beom Seok Han; Kyeong Hun Kim; Daehan Park

doi:10.3934/dcds.2021002

A weighted sobolev space theory for the diffusion-wave equations with time-fractional derivatives on C¹ domains

Beom Seok Han, Kyeong Hun Kim, Daehan Park

Department of Mathematics

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

Abstract

We introduce a weighted Lp-theory (p > 1) for the time-fractional diffusion-wave equation of the type ∂_t^αu(t, x) = a^ij(t, x)u_xi_xj (t, x) + f(t, x), t > 0, x ∈ Ω, where α ∈ (0, 2), ∂_t^α denotes the Caputo fractional derivative of order α, and Ω is a C¹ domain in R^d. We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.

Original language	English
Pages (from-to)	3415-3445
Number of pages	31
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	41
Issue number	7
DOIs	https://doi.org/10.3934/dcds.2021002
Publication status	Published - 2021 Jul

Keywords

Caputo fractional derivative
Domains domains
Sobolev space with weights
Time-fractional equation
Variable coefficients

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.3934/dcds.2021002

Cite this

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title = "A weighted sobolev space theory for the diffusion-wave equations with time-fractional derivatives on C1 domains",

abstract = "We introduce a weighted Lp-theory (p > 1) for the time-fractional diffusion-wave equation of the type ∂tαu(t, x) = aij(t, x)uxixj (t, x) + f(t, x), t > 0, x ∈ Ω, where α ∈ (0, 2), ∂tα denotes the Caputo fractional derivative of order α, and Ω is a C1 domain in Rd. We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.",

keywords = "Caputo fractional derivative, Domains domains, Sobolev space with weights, Time-fractional equation, Variable coefficients",

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

note = "Funding Information: 2020 Mathematics Subject Classification. 45D05, 45K05, 45N05, 35B65, 26A33. Key words and phrases. Time-fractional equation, Caputo fractional derivative, Sobolev space with weights, variable coefficients, C1 domains. The authors were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1A5A1028324). ∗ Corresponding author. Publisher Copyright: {\textcopyright} 2021 American Institute of Mathematical Sciences. All rights reserved.",

year = "2021",

month = jul,

doi = "10.3934/dcds.2021002",

language = "English",

volume = "41",

pages = "3415--3445",

journal = "Discrete and Continuous Dynamical Systems",

issn = "1078-0947",

publisher = "Southwest Missouri State University",

number = "7",

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AU - Han, Beom Seok

AU - Kim, Kyeong Hun

AU - Park, Daehan

N1 - Funding Information: 2020 Mathematics Subject Classification. 45D05, 45K05, 45N05, 35B65, 26A33. Key words and phrases. Time-fractional equation, Caputo fractional derivative, Sobolev space with weights, variable coefficients, C1 domains. The authors were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1A5A1028324). ∗ Corresponding author. Publisher Copyright: © 2021 American Institute of Mathematical Sciences. All rights reserved.

PY - 2021/7

Y1 - 2021/7

N2 - We introduce a weighted Lp-theory (p > 1) for the time-fractional diffusion-wave equation of the type ∂tαu(t, x) = aij(t, x)uxixj (t, x) + f(t, x), t > 0, x ∈ Ω, where α ∈ (0, 2), ∂tα denotes the Caputo fractional derivative of order α, and Ω is a C1 domain in Rd. We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.

AB - We introduce a weighted Lp-theory (p > 1) for the time-fractional diffusion-wave equation of the type ∂tαu(t, x) = aij(t, x)uxixj (t, x) + f(t, x), t > 0, x ∈ Ω, where α ∈ (0, 2), ∂tα denotes the Caputo fractional derivative of order α, and Ω is a C1 domain in Rd. We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.

KW - Caputo fractional derivative

KW - Domains domains

KW - Sobolev space with weights

KW - Time-fractional equation

KW - Variable coefficients

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