Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions

Steady‐state solutions of the aerotaxis problem

“…Hence, |W| ∞ is bounded by a constant independent of p. By the assumption lim p→∞ E ′ (p) = 0 so that there exists p(Ω, E) such that for any two solutions of (15) W 1 , W 2 , and p > p, we have L(W 1 + p, W 2 + p) < 1∕MC 1 (p * , |Ω|). The inequality (20) takes the form…”

“…Global existence and asymptotic behavior of solutions of the parabolic-elliptic counterpart of (8) were studied by Fuest et al 18 Lee et al 19 showed the existence and uniqueness of the steady state of (8) with the Dirichlet boundary condition (4); they also analyzed the behavior of its boundary profile. Hong and Wang 20 studied global solvability and asymptotic behavior of the one-dimensional version of ( 8) with (4). Recently, Lankeit and Winkler 21 analyzed the global solvability of (8) in radial symmetry and with the Dirichlet boundary condition (4).…”

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

“…Global existence of solutions to (1.3) in one-dimensional domains for initial data close to steady states has been investigated in [15], where also local asymptotic stability of the stationary solutions has been shown. The methods of the proofs in [15], which rely on a study of the first primitives of u and v, unfortunately seem inherently restricted to the one-dimensional case.…”

Stability of Steady-State Solutions of a Class of Keller-Segel Models With Mixed Boundary Conditions

“…When γ = 1, m = 1 and φ(w) = w, Braukhoff and Lankeit in [4] proved the stationary solution under the noflux boundary conditions for u and the physically meaningful condition for w in bounded domain Ω ⊂ R n , n ≥ 1. Later, Lee et.al [20] proved the existence and uniqueness of the steady-state solution of equations (1.1)-(1.2) in space Ω ⊂ R n (n ≥ 1); in [35], Winkler consider the existence of globally weak solutions in one-dimensional bounded interval; later, Hong and Wang [13] improved the result of [20] to obtain the asymptotic stability of the steady-state solution in the bounded domain Ω = (0, 1). When γ > 1, m = 0, we refer the readers to [36,12] where the global existence and large time behavior of the one dimensional Cauchy problem solution of equation and dynamic boundary condition problem were proved, respectively.…”

“…However, the stability of steady-state solution to (1.1)-(1.2) with linear sensitivity function is a open problem, which has more biological significance. Motivated by the ideas of [20] and [13], we are devoted to investigating the following system of reaction-diffusion-advection equations:…”