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

Consider the quasilinear diffusion problem $$\{\begin{array}{ll}{\mathbf{u}}^{\mathrm{\prime }}+\mathrm{\Pi }(t,x,\mathbf{u},\mathrm{\Sigma }\mathbf{u})\mathbb{A}\mathbf{u}=\mathbf{f}(t,x,\mathbf{u},\mathrm{\Sigma }\mathbf{u})& \text{ in }]0,T[\times \mathrm{\Omega },\\ \mathbf{u}=\mathbf{0}& \text{ in }]0,T[\times {\mathrm{\Omega }}^c,\\ \mathbf{u}(0,\cdot )={\mathbf{u}}_0(\cdot )& \text{ in }\mathrm{\Omega }\end{array}$$ for an open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, ${\mathbf{u}}_0\in {\mathbf{H}}_0^s(\mathrm{\Omega }):=[H_0^s(\mathrm{\Omega }){]}^m$ and any $T\in ]0,\mathrm{\infty }[$, where $\mathrm{\Sigma }\mathbf{u}\in {\mathbb{R}}^q$ for $0<q\le m\times n$ represents fractional or nonlocal derivatives with order $\sigma$ with $\sigma <2s$ for all $0<s\le 1$, 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 $$(\mathbb{A}\mathbf{u}{)}^i=-\sum _{\alpha ,\beta ,j}{\mathrm{\partial }}_{\alpha }(A_{ij}^{\alpha \beta }{\mathrm{\partial }}_{\beta }{u}^j),\mathbb{A}u=-D^s(A(x)D^{s}u),\text{ and }(\mathbb{A}\mathbf{u}{)}^i={\int }_{{\mathbb{R}}^n}{A}_{ij}(x,y)\frac{u^j(x)-u^j(y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+2s}}dy,$$ for coercive, invertible matrices $\mathrm{\Pi }$ and suitable vectorial functions $\mathbf{f}$.