TY - JOUR

T1 - Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity

AU - Fujie, Kentarou

AU - Winkler, Michael

AU - Yokota, Tomomi

N1 - Funding Information:
T. Yokota is supported by Grant-in-Aid for Scientific Research (C) (No. 25400119 ), JSPS . This work was completed while M. Winkler visited Tokyo University of Science in 2014. He is grateful for the warm hospitality.

PY - 2014/11

Y1 - 2014/11

N2 - This paper is concerned with the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic source,{ut=Δu-χ(u/ vv)+ru-μu2,xεΩ,t>0,0=Δv-v+u,xεΩ, t>0, under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊃ R2, where χ>0,rεR,μ>0, with nonnegative initial data 0≢ u0εC0(Ω ̄). It is shown that in this two-dimensional setting, the absorptive character of the logistic kinetics is sufficient to enforce global existence of classical solutions even for arbitrarily large χ>0 and any μ>0 and rεR. It is moreover shown that if in addition r>0 is sufficiently large then all these solutions are uniformly bounded. A main step in the derivation of these results consists of establishing appropriate positive a priori bounds from below for the mass functional ℓΩu, which due to the presence of logistic kinetics is not preserved. These in turn provide pointwise lower bounds for v, which then allow for the choice of p>1, explicitly depending inter alia on infv, such that ℓΩu p(x,t)dx can be suitably bounded from above.

AB - This paper is concerned with the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic source,{ut=Δu-χ(u/ vv)+ru-μu2,xεΩ,t>0,0=Δv-v+u,xεΩ, t>0, under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊃ R2, where χ>0,rεR,μ>0, with nonnegative initial data 0≢ u0εC0(Ω ̄). It is shown that in this two-dimensional setting, the absorptive character of the logistic kinetics is sufficient to enforce global existence of classical solutions even for arbitrarily large χ>0 and any μ>0 and rεR. It is moreover shown that if in addition r>0 is sufficiently large then all these solutions are uniformly bounded. A main step in the derivation of these results consists of establishing appropriate positive a priori bounds from below for the mass functional ℓΩu, which due to the presence of logistic kinetics is not preserved. These in turn provide pointwise lower bounds for v, which then allow for the choice of p>1, explicitly depending inter alia on infv, such that ℓΩu p(x,t)dx can be suitably bounded from above.

KW - Boundedness

KW - Chemotaxis

KW - Global existence

KW - Logarithmic sensitivity

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U2 - 10.1016/j.na.2014.06.017

DO - 10.1016/j.na.2014.06.017

M3 - Article

AN - SCOPUS:84904963573

VL - 109

SP - 56

EP - 71

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -