TY - JOUR

T1 - How far does small chemotactic interaction perturb the Fisher–KPP dynamics?

AU - Lankeit, Johannes

AU - Mizukami, Masaaki

N1 - Funding Information: J. Lankeit acknowledges support of the Deutsche Forschungsgemeinschaft within the project Analysis of chemotactic cross-diffusion in complex frameworks.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - This paper deals with nonnegative solutions of the Neumann initial–boundary value problem for the fully parabolic chemotaxis-growth system, {(uε)t=Δuε−ε∇⋅(uε∇vε)+μuε(1−uε),x∈Ω,t>0,(vε)t=Δvε−vε+uε,x∈Ω,t>0, with positive small parameter ε>0 in a bounded convex domain Ω⊂Rn (n≥1) with smooth boundary. The solutions converge to the solution u to the Fisher–KPP equation as ε→0. It is shown that for all μ>0 and any suitably regular nonnegative initial data (uinit,vinit) there are some constants ε0>0 and C>0 such that supt>0⁡‖uε(⋅,t)−u(⋅,t)‖L∞(Ω)≤Cεforallε∈(0,ε0).

AB - This paper deals with nonnegative solutions of the Neumann initial–boundary value problem for the fully parabolic chemotaxis-growth system, {(uε)t=Δuε−ε∇⋅(uε∇vε)+μuε(1−uε),x∈Ω,t>0,(vε)t=Δvε−vε+uε,x∈Ω,t>0, with positive small parameter ε>0 in a bounded convex domain Ω⊂Rn (n≥1) with smooth boundary. The solutions converge to the solution u to the Fisher–KPP equation as ε→0. It is shown that for all μ>0 and any suitably regular nonnegative initial data (uinit,vinit) there are some constants ε0>0 and C>0 such that supt>0⁡‖uε(⋅,t)−u(⋅,t)‖L∞(Ω)≤Cεforallε∈(0,ε0).

KW - Chemotaxis

KW - Fisher–KPP equation

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85015031637&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1610.07981

DO - 10.48550/arXiv.1610.07981

M3 - Article

AN - SCOPUS:85015031637

VL - 452

SP - 429

EP - 442

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -