Properties of Generalized Degenerate Parabolic Systems

Abstract

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 $t\geq \tau>0$ 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.