Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form

TL;DR

This paper establishes global existence results for a broad class of nonlocal and anisotropic quasilinear diffusion systems in non-divergence form, encompassing fractional, nonlocal, and classical operators with various boundary conditions.

Contribution

It introduces new global existence theorems for nonlocal and anisotropic quasilinear diffusion systems in non-divergence form, covering diverse operators including fractional and nonlocal types.

Findings

Proved global existence for systems with local elliptic operators.

Extended results to anisotropic fractional and nonlocal operators.

Unified framework for various non-divergence form diffusion systems.

Abstract

Consider the quasilinear diffusion problem \[\begin{cases}\mathbf{u}'+\Pi(t,x,\mathbf{u},\Sigma \mathbf{u})\mathbb{A}\mathbf{u}=\mathbf{f}(t,x,\mathbf{u},\Sigma \mathbf{u})&\text{ in }]0,T[\times\Omega,\\\mathbf{u}=\mathbf{0}&\text{ in }]0,T[\times\Omega^c,\\\mathbf{u}(0,\cdot)=\mathbf{u}_0(\cdot)&\text{ in }\Omega\end{cases}\] for an open set $\Omega\subset\mathbb{R}^n$, $\mathbf{u}_0\in \mathbf{H}^s_0(\Omega):=[H^s_0(\Omega)]^m$ and any $T\in]0,\infty[$, where $\Sigma \mathbf{u}\in \mathbb{R}^q$ for $0<q\leq m\times n$ represents fractional or nonlocal derivatives with order $\sigma$ with $\sigma<2s$ 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 \[(\mathbb{A}\mathbf{u})^i=-\sum _{\alpha,\beta,j} \partial_\alpha(A^{\alpha\beta}_{ij}\partial_\beta u^j),\quad \mathbb{A}u=- D^s(A(x)D^su),\quad\text{ and }\quad (\mathbb{A}\mathbf{u})^i=\int_{\mathbb{R}^n}A_{ij}(x,y)\frac{u^j(x)-u^j(y)}{|x-y|^{n+2s}}\,dy,\] for coercive, invertible matrices $\Pi$ and suitable vectorial functions $\mathbf{f}$.