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

TL;DR

This paper develops a maximal regularity theory with Muckenhoupt weights for time-fractional diffusion-wave equations with variable coefficients, providing sharp regularity results and interpolation properties in weighted Sobolev spaces.

Contribution

It introduces a weighted maximal $L_q(L_p)$-regularity framework for fractional diffusion equations with variable coefficients, including complex interpolation of weighted Sobolev spaces.

Findings

Established maximal regularity estimates with Muckenhoupt weights.

Proved complex interpolation of weighted Sobolev spaces.

Derived sharp regularity results based on the regularity of the source term.

Abstract

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 $1<p,q<\infty$, $\gamma\in\mathbb{R}$, and $w_{1}$ and $w_{2}$ are Muckenhoupt weights. This implies that we prove maximal regularity theory, and sharp regularity of solution according to regularity of $f$. To prove our main result, we also proved the complex interpolation of weighted Sobolev spaces, $$ [H^{\gamma_{0}}_{p_{0}}(w_{0}), H^{\gamma_{1}}_{p_{1}}(w_{1})]_{[\theta]} = H^{\gamma}_{p}(w), $$ where $\theta\in (0,1)$, $\gamma_{0},\gamma_{1}\in\mathbb{R}$, $p_{0},p_{1}\in(1,\infty)$, $w_{i}$ ($i=0,1$) are arbitrary $A_{p_{i}}$ weight, and $$ \gamma=(1-\theta)\gamma_{0}+\theta\gamma_{1}, \quad \frac{1}{p}=\frac{1-\theta}{p_{0}} + \frac{\theta}{p_{1}},\quad w^{1/p}=w^{\frac{(1-\theta)}{p_{0}}}_{0}w^{\frac{\theta}{p_{1}}}_{1}.