  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
$\mathbf{f}$.
