Abstract
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.
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