非各向同性非散度型非局部拟线性扩散系统的全局存在性
偏微分方程分析
2024-04-23 v1
摘要
考虑拟线性扩散问题{ u ′ + Π ( t , x , u , Σ u ) A u = f ( t , x , u , Σ u ) in ] 0 , T [ × Ω , u = 0 in ] 0 , T [ × Ω c , u ( 0 , ⋅ ) = u 0 ( ⋅ ) in Ω \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} ⎩ ⎨ ⎧ u ′ + Π ( t , x , u , Σ u ) A u = f ( t , x , u , Σ u ) u = 0 u ( 0 , ⋅ ) = u 0 ( ⋅ ) in ] 0 , T [ × Ω , in ] 0 , T [ × Ω c , in Ω Ω ⊂ R n \Omega\subset\mathbb{R}^n Ω ⊂ R n u 0 ∈ H 0 s ( Ω ) : = [ H 0 s ( Ω ) ] m \mathbf{u}_0\in \mathbf{H}^s_0(\Omega):=[H^s_0(\Omega)]^m u 0 ∈ H 0 s ( Ω ) := [ H 0 s ( Ω ) ] m T ∈ ] 0 , ∞ [ T\in]0,\infty[ T ∈ ] 0 , ∞ [ Σ u ∈ R q \Sigma \mathbf{u}\in \mathbb{R}^q Σ u ∈ R q 0 < q ≤ m × n 0<q\leq m\times n 0 < q ≤ m × n σ \sigma σ 0 < s ≤ 1 0<s\leq1 0 < s ≤ 1 σ < 2 s \sigma<2s σ < 2 s A \mathbb{A} A ( A u ) i = − ∑ α , β , j ∂ α ( A i j α β ∂ β u j ) , A u = − D s ( A ( x ) D s u ) , 以及 ( A u ) i = ∫ R n A i j ( x , y ) u j ( x ) − u j ( y ) ∣ x − y ∣ n + 2 s d y , (\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{ 以及 }\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, ( A u ) i = − α , β , j ∑ ∂ α ( A ij α β ∂ β u j ) , A u = − D s ( A ( x ) D s u ) , 以及 ( A u ) i = ∫ R n A ij ( x , y ) ∣ x − y ∣ n + 2 s u j ( x ) − u j ( y ) d y , Π \Pi Π f \mathbf{f} f
引用
@article{arxiv.2206.11415,
title = {Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form},
author = {Catharine W. K. Lo and José Francisco Rodrigues},
journal= {arXiv preprint arXiv:2206.11415},
year = {2024}
}