Properties of generalized degenerate parabolic systems

Abstract

In this article, we consider the parabolic system (u(i))(t) = del.(mU(m-1) A(del u(i), u(i), x, t) + B(u(i), x, t)), (1 <= i <= k) in the range of exponents > m n-2/n where the diffusion coefficient U depends on the components of the solution u = (u(1), ..., u(k)). Under suitable structure conditions on the vector fields A and B, we first showed the uniform L-infinity boundedness of the functionU for t >= tau > 0. We also proved the law of L-1 mass conservation and the local continuity of solution u. In the last result, all components of the solution u have the same modulus of continuity if the ratio between U and u(i), (1 <= i <= k), is uniformly bounded above and below.