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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2206.11415 |
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| _version_ | 1866913322676256768 |
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| author | Lo, Catharine W. K. Rodrigues, José Francisco |
| author_facet | Lo, Catharine W. K. Rodrigues, José Francisco |
| contents | 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 $Ω\subset\mathbb{R}^n$, $\mathbf{u}_0\in \mathbf{H}^s_0(Ω):=[H^s_0(Ω)]^m$ and any $T\in]0,\infty[$, where $Σ\mathbf{u}\in \mathbb{R}^q$ for $0<q\leq m\times n$ represents fractional or nonlocal derivatives with order $σ$ with $σ<2s$ 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 \[(\mathbb{A}\mathbf{u})^i=-\sum _{α,β,j} \partial_α(A^{αβ}_{ij}\partial_βu^j),\quad \mathbb{A}u=- D^s(A(x)D^su),\quad\text{ and }\quad (\mathbb{A}\mathbf{u})^i=\int_{\mathbb{R}^n}A_{ij}(x,y)\frac{u^j(x)-u^j(y)}{|x-y|^{n+2s}}\,dy,\] for coercive, invertible matrices $Π$ and suitable vectorial functions $\mathbf{f}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_11415 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form Lo, Catharine W. K. Rodrigues, José Francisco Analysis of PDEs 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 $Ω\subset\mathbb{R}^n$, $\mathbf{u}_0\in \mathbf{H}^s_0(Ω):=[H^s_0(Ω)]^m$ and any $T\in]0,\infty[$, where $Σ\mathbf{u}\in \mathbb{R}^q$ for $0<q\leq m\times n$ represents fractional or nonlocal derivatives with order $σ$ with $σ<2s$ 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 \[(\mathbb{A}\mathbf{u})^i=-\sum _{α,β,j} \partial_α(A^{αβ}_{ij}\partial_βu^j),\quad \mathbb{A}u=- D^s(A(x)D^su),\quad\text{ and }\quad (\mathbb{A}\mathbf{u})^i=\int_{\mathbb{R}^n}A_{ij}(x,y)\frac{u^j(x)-u^j(y)}{|x-y|^{n+2s}}\,dy,\] for coercive, invertible matrices $Π$ and suitable vectorial functions $\mathbf{f}$. |
| title | Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2206.11415 |