In this paper, we are concerned with a cross-diffusive evolution system arising from biological transport networks. We study the general regularity properties of solutions to the corresponding Dirichlet-Neumann initial-boundary value problem (DNibvp). By applying the properties of divergence equations and the Dirichlet heat semigroup, we find that DNibvp possesses a globally classical solution which is unique and uniformly bounded. Based on this uniform boundedness, we establish the existence of the steady states by means of dynamical methods. Our results demonstrate that the transient solution will stabilize to the stationary solution in infinite time with an exponential time-decay rate.

在本文中，我们关注由生物运输网络引起的交叉扩散进化系统。我们研究了相应的Dirichlet-Neumann初边值问题（DNibvp）的解的一般正则性质。通过应用散度方程和Dirichlet热半群的性质，我们发现DNibvp拥有唯一且有界的全局经典解。基于这种一致的有界性，我们通过动力学方法确定了稳态的存在。我们的结果表明，瞬态解将在无限时间内以指数衰减率稳定到平稳解。