Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks

Abstract

The aim of this article is to consider a three-dimensional Cauchy problem for the parabolic-elliptic system arising from biological transport networks. For such problem, we first establish the global existence, uniqueness, and uniform boundedness of the strong solution by estimating the derivative of the diagonal permeability tensor with respect to time variable. Moreover, for the diffusion coefficient appropriately large, we demonstrate that the corresponding stationary problem admits a strong solution and that the solution of the Cauchy problem will stabilize to its stationary counterpart in infinite time with a time-decay rate.

Keywords

• parabolic-elliptic system
• global existence
• boundedness
• steady state
• 35k55
• 35a01
• 35b35
• 35b40
• 35q92