A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains

Beom-Seok Han; Kyeong-Hun Kim; Daehan Park

doi:10.3934/dcds.2021002

Article Contents

2021, Volume 41, Issue 7: 3415-3445. Doi: 10.3934/dcds.2021002

This issue Previous Article Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance Next Article Constant-speed ramps for a central force field

A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains

Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea

^* Corresponding author
^* Corresponding author

Received: February 2020

Revised: November 2020

Early access: January 2021

Published: July 2021

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

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Abstract

We introduce a weighted $ L_{p} $-theory ($ p>1 $) for the time-fractional diffusion-wave equation of the type

$ \begin{equation*} \partial_{t}^{\alpha}u(t,x) = a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x), \quad t>0, x\in \Omega, \end{equation*} $

where $ \alpha\in (0,2) $, $ \partial_{t}^{\alpha} $ denotes the Caputo fractional derivative of order $ \alpha $, and $ \Omega $ is a $ C^1 $ domain in $ \mathbb{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.

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Mathematics Subject Classification: 45D05, 45K05, 45N05, 35B65, 26A33.

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