Long time behavior of the solution to a parabolic–elliptic system

Abstract

In this paper, we are concerned with a parabolic–elliptic model for biological fluid transportation networks with the large diffusivity and activation. Our investigations reveal that local existence of the unique classical solution to the correspondinginitial–boundary value problem can always be achieved in the n-dimensional setting with n≤3. Moreover, we show that this local solution can be extended to a global one as long as the source term and the initial data are made small enough in a suitable sense, and that the classical solution converges exponentially to a trivial steady state; meanwhile, we present that the smallness condition on the initial data can be removed in the one-dimensional setting. Our result implies that we cannot expect network formation when the diffusivity is sufficiently large, which is a first step toward a qualitative analysis for the role of diffusion in biological transportation networks.