论文标题
$ L_Q（L_P）$ - 带有时空非本地运算符扩散方程的理论
An $L_q(L_p)$-theory for diffusion equations with space-time nonlocal operators
论文作者
论文摘要
我们提出了一个$ l_q（l_ {p}）$ - 方程$ \ partial_ {t}^αu= ϕ（δ） $$这里$ p，q> 1 $，$α\ in（0,1）$，$ \ partial_ {t}^α$是订单$α$的caputo分数衍生物，$ ϕ $是伯恩斯坦的功能，是满足以下内容的：$ \ evente; \ label {eqn 8.17.1} c \ left（\ frac {r} {r} \ right）^{Δ_0} \ leq \ frac {ϕ（r）} {r）} {x（r）}，\ qquad 0 <qquad 0 <r <r <r <r <r <\ f \ iffty。 \ end {equation}我们证明了唯一性和存在在sobolev空间中产生的结果，并获得了解决方案的最大规律性结果。特别是，我们证明\ begin {align*} \ | | \ \ partial^α_tU |+|+|+| ϕ（δ） αQ}}），\ end {align*}其中$ b_ {p，q}^{ϕ，2-2/αq} $是与$ \ mathbb {r}^d $相关的修改后的besov space。 我们的方法基于$ p = q $的BMO估计，而valuredcalderón-zygmund定理则以$ p \ neq q $为基础。 Littlewood-Paley理论还用于处理非零的初始数据问题。我们的证明依赖于基本解决方案的衍生估计，这是根据概率理论在本文中获得的。
We present an $L_q(L_{p})$-theory for the equation $$ \partial_{t}^αu=ϕ(Δ) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $α\in (0,1)$, $\partial_{t}^α$ is the Caputo fractional derivative of order $α$, and $ϕ$ is a Bernstein function satisfying the following: $\exists δ_0\in (0,1]$ and $c>0$ such that \begin{equation} \label{eqn 8.17.1} c \left(\frac{R}{r}\right)^{δ_0}\leq \frac{ϕ(R)}{ϕ(r)}, \qquad 0<r<R<\infty. \end{equation} We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove \begin{align*} \| |\partial^α_t u|+|u|+|ϕ(Δ)u|\|_{L_q([0,T];L_p)}\leq N(\|f\|_{L_q([0,T];L_p)}+ \|u_0\|_{B_{p,q}^{ϕ,2-2/ αq}}), \end{align*} where $B_{p,q}^{ϕ,2-2/αq}$ is a modified Besov space on $\mathbb{R}^d$ related to $ϕ$. Our approach is based on BMO estimate for $p=q$ and vector-valued Calderón-Zygmund theorem for $p\neq q$. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.
