Convergence In Comparable Almost Periodic Reaction-diffusion Systems With Dirichlet Boundary Conditions

In the previous 50 years or so, many concepts from dynamical systems have been applied to the study of partial differential equations. Especially the monotone dynamical systems have been widely studied, because these systems provide a unified relevant mathematical framework for the qualitative analysis of many important equations. However, a large amount of important evolution equations do not generate monotone systems, In order to study properties of the solutions of such non-monotone evolution equations, an effective approach is to exhibit and utilize certain comparison techniques.it has been turned out that this comparison technique involves monotone systems in a very natural way: the original non-monotone systems are comparable with respect to some monotone systems.In this article, we study the asymptotic dynamics in nonmonotone comparable almost periodic reaction-diffusion systems with Dirichlet boundary condition, which are comparable with uniformly stable strongly order-preserving systems. we first prove that every precompact trajectory of the strongly order-preserving system is asymptotic to a 1-cover of the base flow. Based on this, for the uniformly stable and strongly order-preserving skew-product semiflow, we can get the topological structure of the set of the union of all 1-covers similarly as paper[3]. With such tools, we are able to establish the 1-covering property of uniformly stable omega-limit sets of comparable skew-product semifow, and thus obtain the asymptotic almost periodicity of uniformly stable solutions.