Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity

Zhao, Xiangdong

doi:10.3934/dcdsb.2020334

Cited by 1 publication

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“…For the corresponding parabolic-parabolic system with logistic-type source f(u) = ru − µu γ with γ > 1, the global existence-boundedness of classical or generalized solutions has been constructed in [6,29] by means of the a priori point-in-wise lower bound or uniformly lower bound for v derived by requiring r suitably large than χ, γ like that in (1.3). When replacing the second equation by v t = ϵ∆v − v + u, it is proved in [30] that the system admits a global bounded classical solutions if χ + ϵ ∈ (0, 1) for arbitrary r > 0 and γ > 1 without the restriction on the large r or strong damping exponent γ.…”

Section: Introductionmentioning

confidence: 99%

Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two

Zhao

2023

Nonlinearity

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This paper deals with the parabolic-elliptic Keller–Segel system with weakly singular sensitivity and logistic source under the homogeneous Neumann boundary: u t = Δ u − χ ∇ ⋅ ( u v α ∇ v ) + r u − μ u 2 , 0 = Δ v − v + u in a smooth bounded domain Ω ⊂ R 2 , with χ , α , μ > 0 , r ∈ R . It is proved that the system possesses a globally bounded classical solution for α ∈ ( 0 , 1 ) with µ > 0 suitably large, without establishing the uniformly positive bound for v from below. In addition, we give the explicit expression of the upper bound for solution u with respect to the parameters χ , α , r , μ via a recursive argument on α. This concludes that weakly singular sensitivity benefits to obtain the global boundedness of classical solution in dimension two.

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Section: Introductionmentioning

confidence: 99%