Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
184 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Weighted maximal $L_{q}(L_{p})$-regularity theory for time-fractional diffusion-wave equations with variable coefficients (2103.13673v2)

Published 25 Mar 2021 in math.AP

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}.

Summary

We haven't generated a summary for this paper yet.