Properties of generalized degenerate parabolic systems

Abstract

In this article, we consider the parabolic system (ui)t=∇⋅(mUm−1A(∇ui,ui,x,t)+ℬ(ui,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−2nm\gt \frac{n-2}{n} where the diffusion coefficient UU depends on the components of the solution u=(u1,…,uk){\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 UU for t≥τ>0t\ge \tau \gt 0. We also proved the law of L1{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 UU and ui{u}^{i}, (1≤i≤k1\le i\le k), is uniformly bounded above and below.

Keywords

• local continuity of degenerate parabolic systems
• uniform boundedness
• law of mass conservation
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