Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension

Abstract

This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.

Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis

“…We refer to [7,26,27] for the global existence and large-time behavior of solutions to (1.6) and [28,30] for some foraging models which have similar consumption terms as in (1.6). When the second equation of (1.6) is replaced by an elliptic equation ǫ∆u + v = 0, the radially symmetric boundary-layer solution as ǫ → 0 has been studied in [3]. If the rst equation of (1.6) is replaced by the v t = ∆v − ∇ • (v∇ log u), namely the chemotactic sensitivity is logarithmic, and the Dirichlet boundary condition for v and Robin boundary condition for u are prescribed, the boundary-layer solution of time-dependent problem has been studied in a series works [11][12][13] where the boundary-layer appears in the gradient of u other than u itself.…”