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

Abstract

We introduce a weighted Lp-theory (p > 1) for the time-fractional diffusion-wave equation of the typet denotes the Caputo fractional derivative of order α, and Ω is a C 1 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.

L-estimates for time fractional parabolic equations with coefficients measurable in time

“…To provide context for our work on time fractional parabolic equations and related results in the literature, we refer the reader to the paper [9] and the references therein. Also see [12], where the authors deal with equations similar to those in this paper but in a different type of weighted Sobolev spaces with α ∈ (0, 2) and continuous a ij (t, x). Further, one can find related results on time fractional evolution equations in Hilbert space settings in [18,17,16,1].…”

Determinants of Monthly Rent in Small and Medium Office Buildings: Focusing on the cases of Gangnam-gu, Songpa-gu, and Seocho-gu in Seoul

“…Finally, we mention that the approach presented in this paper can be applied not only to the Poisson and heat equations but also to extended evolution equations, such as the time-fractional heat equations and the stochastic heat equation (for definitions, see, e.g., [26,34] and [35,39,44], respectively). The localization argument presented in Section 4 and the results that provide appropriate superharmonic functions for each domain (see Sections 5 and 6) can be directly applied to these equations.…”