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

Abstract

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 $\Omega \subset {\mathbb{R}}^n$, ${\mathbf{u}}_0\in {\mathbf{H}}_0^s(\Omega ):=[H_0^s(\Omega ){]}^m$ and any $T\in ]0,\infty [$, where $\Sigma \mathbf{u}\in {\mathbb{R}}^q$ 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 \[(\mathbb{A}\mathbf{u})^i=-\sum _{α,β,j} \partial_α(A^{αβ}_{ij}\partial_βu^j),\quad \mathbb{A}u=- D^s(A(x)D^su),\quad\text{ and }\quad (\mathbb{A}\mathbf{u})^i=\int_{\mathbb{R}^n}A_{ij}(x,y)\frac{u^j(x)-u^j(y)}{|x-y|^{n+2s}}\,dy,\] for coercive, invertible matrices $\Pi$ and suitable vectorial functions $\mathbf{f}$.

References in corpus (1)

• On a Class of Nonlocal Obstacle Type Problems Related to the Distributional Riesz Fractional Derivative

Cited by in corpus (2)

• On inverse problems in multi-population aggregation models
• Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems