TY - JOUR

T1 - A weighted sobolev space theory for the diffusion-wave equations with time-fractional derivatives on C1 domains

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

UR - http://www.scopus.com/inward/record.url?scp=85103787926&partnerID=8YFLogxK

U2 - 10.3934/dcds.2021002

DO - 10.3934/dcds.2021002

M3 - Article

AN - SCOPUS:85103787926

VL - 41

SP - 3415

EP - 3445

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 7

ER -