Properties of generalized degenerate parabolic systems

Abstract

Abstract In this article, we consider the parabolic system ( u i ) t = ∇ ⋅ ( m U m − 1 A ( ∇ u i , u i , x , t ) + ℬ ( 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 ℬ {\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.