Large time behavior of solutions to a cooperative model with population flux by attractive transition

Abstract

This paper is concerned with a diffusive Lotka-Volterra cooperative modelwith population flux by attractive transition. We study the time-global well-posedness and the large time behavior of solutions in a case where the habitat is a bounded convex domain and random diffusion rates equal to each other. A main result shows that when the spatial dimension is less than or equal to 3, under a weaker cooperative condition, a classical solution exists globally in time if the initial data belongs to a suitable functional space. Furthermore, it is shown that if there exists a positive steady state and the equal random diffusion rate is sufficiently large, then all positive solutions asymptotically approach the positive steady state as time tends to infinity.