TY - JOUR
T1 - Global well-posedness and asymptotic stabilization for chemotaxis system with signal-dependent sensitivity
AU - Ahn, Jaewook
N1 - Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2019/5/5
Y1 - 2019/5/5
N2 - A fully parabolic chemotaxis system u t =Δu−∇⋅(uχ(v)∇v),v t =Δv−v+u, in a smooth bounded domain Ω⊂R N , N≥2 with homogeneous Neumann boundary conditions is considered, where the non-negative chemotactic sensitivity function χ satisfies χ(v)≤μ(a+v) −k , for some a≥0 and k≥1. It is shown that a novel type of weight function can be applied to a weighted energy estimate for k>1. Consequently, the range of μ for the global existence and uniform boundedness of classical solutions established by Mizukami and Yokota [23] is enlarged. Moreover, under a convexity assumption on Ω, an asymptotic Lyapunov functional is obtained and used to establish the asymptotic stability of spatially homogeneous equilibrium solutions for k≥1 under a smallness assumption on μ. In particular, when χ(v)=μ/v and N<8, it is shown that the spatially homogeneous steady state is a global attractor whenever μ≤1/2.
AB - A fully parabolic chemotaxis system u t =Δu−∇⋅(uχ(v)∇v),v t =Δv−v+u, in a smooth bounded domain Ω⊂R N , N≥2 with homogeneous Neumann boundary conditions is considered, where the non-negative chemotactic sensitivity function χ satisfies χ(v)≤μ(a+v) −k , for some a≥0 and k≥1. It is shown that a novel type of weight function can be applied to a weighted energy estimate for k>1. Consequently, the range of μ for the global existence and uniform boundedness of classical solutions established by Mizukami and Yokota [23] is enlarged. Moreover, under a convexity assumption on Ω, an asymptotic Lyapunov functional is obtained and used to establish the asymptotic stability of spatially homogeneous equilibrium solutions for k≥1 under a smallness assumption on μ. In particular, when χ(v)=μ/v and N<8, it is shown that the spatially homogeneous steady state is a global attractor whenever μ≤1/2.
KW - Chemotaxis
KW - Global existence
KW - Stabilization
KW - Weight function
UR - http://www.scopus.com/inward/record.url?scp=85056466918&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2018.11.015
DO - 10.1016/j.jde.2018.11.015
M3 - Article
AN - SCOPUS:85056466918
SN - 0022-0396
VL - 266
SP - 6866
EP - 6904
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 10
ER -