论文标题
非局限性准扩散系统的全球存在，以非各向同性非差异形式
Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form
论文作者
论文摘要
考虑准线性扩散问题\ [\ begin {case} \ Mathbf {u}'+π（t，x，x，\ mathbf {u}，σ\ mathbf {u}）\ Math bb {a} \ mathbf {u} = \ mathbf {f}（t，x，x，x，\ mathbf {u}，σ\ \ mathbf {u}）＆\ text { in}] 0，t [\timesΩ，\\\ mathbf {u} = \ mathbf {0}＆\ text {in}] 0，t [\ timesmom^c，\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ cdot）开放设置$ω\ subset \ mathbb {r}^n $，$ \ \ \ m mathbf {u} _0 \ in \ mathbf {h}^s_0（ω） $σ\ MathBf {u} \ in \ Mathbb {r}^q $对于$ 0 <q \ leq m \ leq m \ times n $表示分数或非局部衍生物，其订单$σ$，$σ<2s $，全部$ 0 <s \ leq1 $，包括经典级别的订单和衍生的序列范围，包括$ 0 <s \ s $ quas，包括各种序列的序列和衍生的序列范围。对于不同的线性操作员$ \ mathbb {a} $，包括局部椭圆系统，各向异性分数方程和系统以及各向异性非局限性运算符，以下类型\ [（\ Mathbb {a} {a} \ Mathbf {u} i = - \partial_α（a^{αβ} _ {ij} \partial_βu^j），\ quad \ mathbb {a} u = - d^s（a（x）d^su），\ quad \ quad \ text \ text {and} \ quad {and} \ quad （\ Mathbb {a} \ MathBf {u}）^i = \ int _ {\ MathBb {r}^n} a_ {ij}（x，x，x，y）\ frac {u^j（x）-u^J（x）-U^J（y）-u^J（y）} $π$和合适的矢量函数$ \ mathbf {f} $。
Consider the quasilinear diffusion problem \[\begin{cases}\mathbf{u}'+Π(t,x,\mathbf{u},Σ\mathbf{u})\mathbb{A}\mathbf{u}=\mathbf{f}(t,x,\mathbf{u},Σ\mathbf{u})&\text{ in }]0,T[\timesΩ,\\\mathbf{u}=\mathbf{0}&\text{ in }]0,T[\timesΩ^c,\\\mathbf{u}(0,\cdot)=\mathbf{u}_0(\cdot)&\text{ in }Ω\end{cases}\] for an open set $Ω\subset\mathbb{R}^n$, $\mathbf{u}_0\in \mathbf{H}^s_0(Ω):=[H^s_0(Ω)]^m$ and any $T\in]0,\infty[$, where $Σ\mathbf{u}\in \mathbb{R}^q$ for $0<q\leq m\times n$ represents fractional or nonlocal derivatives with order $σ$ with $σ<2s$ for all $0<s\leq1$, including the classical gradient and derivatives of order greater than 1. We show global existence results for various quasilinear diffusion systems in non-divergence form, for different linear operators $\mathbb{A}$, including local elliptic systems, anisotropic fractional equations and systems, and anisotropic nonlocal operators, of the following type \[(\mathbb{A}\mathbf{u})^i=-\sum _{α,β,j} \partial_α(A^{αβ}_{ij}\partial_βu^j),\quad \mathbb{A}u=- D^s(A(x)D^su),\quad\text{ and }\quad (\mathbb{A}\mathbf{u})^i=\int_{\mathbb{R}^n}A_{ij}(x,y)\frac{u^j(x)-u^j(y)}{|x-y|^{n+2s}}\,dy,\] for coercive, invertible matrices $Π$ and suitable vectorial functions $\mathbf{f}$.
