论文标题
广义退化抛物线系统的特性
Properties of Generalized Degenerate Parabolic Systems
论文作者
论文摘要
在本文中，我们考虑了通用抛物线抛物面系统\ begin \ begin \ begin {equation*} \ left（u^i^i \ right）_t = \ nabla \ nabla \ nabla \ cdot \ cdot \ cdot \ cdot \ lest（m-1} \ nabl（m-nabl）， u^i，u^i，x，x，t \ right）+\ mathcal {b} \ left（u^i，x，x，t \ right）\ right），\ qquad \ left（1 \ leq i \ leq k \ right）取决于解决方案$ \ bold {u} $的组件。在矢量字段的适当结构条件下，$ \ MATHCAL {a} $和$ \ MATHCAL {b} $，我们首先显示函数$ u $ bond的均匀$ l^{\ infty} $，用于$ t \ geq geq geq> 0 $，$ l^of $ l^1 $ $ l^1 $ $ l^1 $ systone $ u^i $ u^i $ u^i $（c）不平等。作为最后的结果，我们还处理了解决方案的本地连续性$ \ bold {u} = \ left（u^1，\ cdots，u^k \ right）$，并使用固有的缩放。如果$ u $和组件之间的比率$ u^i $，$（i = 1，\ cdots，k）$在上方和下方均匀地界限，则解决方案$ \ bold {u} $的所有组件都具有相同的连续性模量。
In this paper, we consider the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ of the generalized parabolic system \begin{equation*} \left(u^i\right)_t=\nabla\cdot\left(mU^{m-1}\mathcal{A}\left(\nabla u^i,u^i,x,t\right)+\mathcal{B}\left(u^i,x,t\right)\right), \qquad \left(1\leq i\leq k\right) \end{equation*} in the range of exponents $m>\frac{n-2}{n}$ where the diffusion coefficient $U$ depends on the components of the solution $\bold{u}$. Under suitable structure conditions on the vector fields $\mathcal{A}$ and $\mathcal{B}$, we first show the uniform $L^{\infty}$ bound of the function $U$ for $t\geq τ>0$ and law of $L^1$ mass conservation of each component $u^i$, $(i=1,\cdots,k)$, with system version of Harnack type inequality. As the last result, we also deal with the local continuity of solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ with the intrinsic scaling. If the ratio between $U$ and components $u^i$, $(i=1,\cdots,k)$, is uniformly bounded above and below, all components of the solution $\bold{u}$ have the same modulus of continuity.
