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

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}.