Author’s summary: ”We consider mainly a system of reaction-diffusion equations with general diffusion matrix and we establish the stabilization of all solutions at \(t\to \infty\). The interest of this problem derives from two separate facts. First, the sets that are useful for localizing the asymptotics cease to be invariant as soon as the diffusion matrix is not a multiple of the identity. Second, the set of equilibria is connected. Further, we establish uniform \(L^{\infty}\) bounds for the solutions of a class of parabolic systems. The unifying feature in the problems considered is the lack of any conventional maximum principles.”