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Properties of generalized degenerate parabolic systems
Advances in Nonlinear Analysis ( IF 3.2 ) Pub Date : 2022-03-09 , DOI: 10.1515/anona-2022-0236
Sunghoon Kim 1 , Ki-Ahm Lee 2
Affiliation  

In this article, we consider the parabolic system ( u i ) t = ( m U m 1 A ( u i , u i , x , t ) + ( u i , x , t ) ) , ( 1 i k ) {({u}^{i})}_{t}=\nabla \cdot (m{U}^{m-1}{\mathcal{A}}(\nabla {u}^{i},{u}^{i},x,t)+{\mathcal{ {\mathcal B} }}({u}^{i},x,t)),\hspace{1.0em}(1\le i\le k) in the range of exponents m > n 2 n m\gt \frac{n-2}{n} where the diffusion coefficient U U depends on the components of the solution u = ( u 1 , , u k ) {\bf{u}}=({u}^{1},\ldots ,{u}^{k}) . Under suitable structure conditions on the vector fields A {\mathcal{A}} and {\mathcal{ {\mathcal B} }} , we first showed the uniform L {L}^{\infty } boundedness of the function U U for t τ > 0 t\ge \tau \gt 0 . We also proved the law of L 1 {L}^{1} mass conservation and the local continuity of solution u {\bf{u}} . In the last result, all components of the solution u {\bf{u}} have the same modulus of continuity if the ratio between U U and u i {u}^{i} , ( 1 i k 1\le i\le k ), is uniformly bounded above and below.

中文翻译:

广义退化抛物线系统的性质

在本文中,我们考虑抛物线系统 ( 一世 ) = ( ü - 1 一种 ( 一世 , 一世 , X , ) + ( 一世 , X , ) ) , ( 1 一世 ķ ) {({u}^{i})}_{t}=\nabla \cdot (m{U}^{m-1}{\mathcal{A}}(\nabla {u}^{i},{ u}^{i},x,t)+{\mathcal{ {\mathcal B} }}({u}^{i},x,t)),\hspace{1.0em}(1\le i\乐 k) 在指数范围内 > n - 2 n m\gt \frac{n-2}{n} 其中扩散系数 ü ü 取决于解决方案的组成部分 = ( 1 , , ķ ) {\bf{u}}=({u}^{1},\ldots ,{u}^{k}) . 在合适的结构条件下的矢量场 一种 {\数学{A}} {\mathcal{ {\mathcal B} }} ,我们首先展示了制服 大号 {L}^{\infty } 函数的有界 ü ü 为了 τ > 0 t\ge\tau\gt 0 . 我们还证明了 大号 1 {L}^{1} 质量守恒和解的局部连续性 {\bf{u}} . 在最后的结果中,解决方案的所有组件 {\bf{u}} 具有相同的连续模量,如果之间的比率 ü ü 一世 {u}^{i} , ( 1 一世 ķ 1\le i\le k ),上下一致。
更新日期:2022-03-09
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