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

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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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  • 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.

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


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