TY - JOUR

T1 - Application of an Adams type inequality to a two-chemical substances chemotaxis system

AU - Fujie, Kentarou

AU - Senba, Takasi

N1 - Funding Information:
The authors would like to express their gratitude to the referee for kind advice. The second author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 26400172), Japan Society for the Promotion of Science.
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2017/7/5

Y1 - 2017/7/5

N2 - This paper deals with positive solutions of the fully parabolic system, {ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞), under homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂Rn (n≤4) with positive parameters τ1,τ2,χ<0 and nonnegative smooth initial data (u0,v0,w0). In the lower dimensional case (n≤3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded. In the case n=4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if ‖u0‖L1(Ω)>(8π)2/χ; as to the mixed boundary value problem, we will establish global existence and boundedness of solutions if ‖u0‖L1(Ω)>(8π)2/χ without radial symmetry. The key ingredients are a Lyapunov functional and an Adams type inequality. A Lyapunov functional of the above problems will be constructed and the constant (8π)2/χ is deduced from the critical constant in the Adams type inequality. This result is regarded as a generalization of the well-known 8π problem in the Keller–Segel system to higher dimensions.

AB - This paper deals with positive solutions of the fully parabolic system, {ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞), under homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂Rn (n≤4) with positive parameters τ1,τ2,χ<0 and nonnegative smooth initial data (u0,v0,w0). In the lower dimensional case (n≤3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded. In the case n=4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if ‖u0‖L1(Ω)>(8π)2/χ; as to the mixed boundary value problem, we will establish global existence and boundedness of solutions if ‖u0‖L1(Ω)>(8π)2/χ without radial symmetry. The key ingredients are a Lyapunov functional and an Adams type inequality. A Lyapunov functional of the above problems will be constructed and the constant (8π)2/χ is deduced from the critical constant in the Adams type inequality. This result is regarded as a generalization of the well-known 8π problem in the Keller–Segel system to higher dimensions.

KW - Adams' inequality

KW - Chemotaxis

KW - Global existence

KW - Lyapunov functional

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U2 - 10.1016/j.jde.2017.02.031

DO - 10.1016/j.jde.2017.02.031

M3 - Article

AN - SCOPUS:85014444707

VL - 263

SP - 88

EP - 148

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -