Weighted maximal \(L_{q}(L_{p})\)-regularity theory for time-fractional diffusion-wave equations with variable coefficients

We present a maximal \(L_{q}(L_{p})\)-regularity theory with Muckenhoupt weights for the equation \begin{equation}\label{eqn 01.26.16:00} \partial^{\alpha}_{t}u(t,x)=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad t>0,x\in\mathbb{R}^{d}. \end{equation} Here, \(\partial^{\alpha}_{t}\) is the Caputo fr...

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Bibliographic Details
Published in:arXiv.org 2022-11
Main Author: Park, Daehan
Format: Article
Language:English
Subjects:
Interpolation
Regularity
Sobolev space
Wave equations
Online Access:Get full text
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Summary:We present a maximal \(L_{q}(L_{p})\)-regularity theory with Muckenhoupt weights for the equation \begin{equation}\label{eqn 01.26.16:00} \partial^{\alpha}_{t}u(t,x)=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad t>0,x\in\mathbb{R}^{d}. \end{equation} Here, \(\partial^{\alpha}_{t}\) is the Caputo fractional derivative of order \(\alpha\in(0,2)\) and \(a^{ij}\) are functions of \((t,x)\). Precisely, we show that \begin{equation*} \begin{aligned} &\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-\Delta)^{\gamma/2}u_{xx}(t,x)|^{p}w_{1}(x)dx\right)^{q/p}w_{2}(t)dt \\ &\quad \leq N \int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-\Delta)^{\gamma/2}f(t,x)|^{p}w_{1}(x)dx\right)^{q/p}w_{2}(t)dt, \end{aligned} \end{equation*} where \(1
ISSN:2331-8422