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

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 $\mathbf{f}$.