TY - JOUR

T1 - Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions

AU - Fujie, Kentaro

AU - Senba, Takasi

N1 - Funding Information:
The authors thank the anonymous referee’s careful reading and useful suggestions. K Fujie is supported by Japan Society for the Promotion of Science (Grant-in-Aid for Early-Career Scientists; No. 19K14576). T Senba is supported by Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research(C); No. 18K03386)
Publisher Copyright:
© 2022 IOP Publishing Ltd & London Mathematical Society.

PY - 2022/7/7

Y1 - 2022/7/7

N2 - This paper deals with classical solutions to the parabolic-parabolic system ut=Δ(Δ(v)u)inω×(0,∞),vt=Δv-v+uinω×(0,∞),∂u∂ν=∂v∂ν=0on∂ω×(0,∞),u(·,0)=u0,v(·,0)=v0inω, where ω is a smooth bounded domain in R n (n ∼ 3), Δ(v) = v -k (k > 0) and the initial data (u 0, v 0) is positive and regular. This system has striking features similar to those of the logarithmic Keller-Segel system. It is established that classical solutions of the system exist globally in time and remain uniformly bounded in time if k ϵ(0, n/(n - 2)), independently of the magnitude of mass. This constant n/(n - 2) is conjectured as the optimal range guaranteeing global existence and boundedness in the corresponding logarithmic Keller-Segel system. In the course of our analysis we introduce an auxiliary function and derive an evolution equation that it satisfies. Using refined comparison estimates we control the behavior of the nonlinear term in the equation for the auxiliary function, and this in turn yields sufficient information to control solutions of the original system.

AB - This paper deals with classical solutions to the parabolic-parabolic system ut=Δ(Δ(v)u)inω×(0,∞),vt=Δv-v+uinω×(0,∞),∂u∂ν=∂v∂ν=0on∂ω×(0,∞),u(·,0)=u0,v(·,0)=v0inω, where ω is a smooth bounded domain in R n (n ∼ 3), Δ(v) = v -k (k > 0) and the initial data (u 0, v 0) is positive and regular. This system has striking features similar to those of the logarithmic Keller-Segel system. It is established that classical solutions of the system exist globally in time and remain uniformly bounded in time if k ϵ(0, n/(n - 2)), independently of the magnitude of mass. This constant n/(n - 2) is conjectured as the optimal range guaranteeing global existence and boundedness in the corresponding logarithmic Keller-Segel system. In the course of our analysis we introduce an auxiliary function and derive an evolution equation that it satisfies. Using refined comparison estimates we control the behavior of the nonlinear term in the equation for the auxiliary function, and this in turn yields sufficient information to control solutions of the original system.

KW - 35B45

KW - 35K57

KW - 35Q92

KW - 92C17

KW - chemotaxis

KW - global existence

KW - Keller-Segel system

KW - sensitivity function

KW - uniform boundedness

UR - http://www.scopus.com/inward/record.url?scp=85133608885&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85133608885&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/ac6659

DO - 10.1088/1361-6544/ac6659

M3 - Article

AN - SCOPUS:85133608885

SN - 0951-7715

VL - 35

SP - 3777

EP - 3811

JO - Nonlinearity

JF - Nonlinearity

IS - 7

ER -