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Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2019-11-26 , DOI: 10.1016/j.jde.2019.11.052
Mengyao Ding , Wei Wang , Shulin Zhou , Sining Zheng

In this paper we study the asymptotic behaviors of global solutions to the fully parabolic chemotaxis system: ut=(D(u)uS(u)v)+ruμu1+σ, vt=Δvv+uγ, subject to the homogeneous Neumann boundary conditions in a bounded and smooth domain ΩRn (n2), where parameters μ,σ,γ>0, rR, and the nonlinearity D,SC2([0,)) are supposed to generalize the prototypesD(u)a0(u+1)α,0S(u)b0u(u+1)β1 with a0,b0>0 and α,βR. We first consider the case of r>0 and provide a boundedness result under α+β+γ<2n, or β+γ<1+σ, or β+γ=1+σ with large μ>0. The main result is concerned with the asymptotic stability when damping effects of logistic source are strong enough. Specifically, there is μ0>0 independent of initial data, such that the bounded classical solution (u,v) satisfies (u,v)((rμ)1σ,(rμ)γσ) in L(Ω) exponentially under conditions of μ>μ0 and r>0. For the case of r<0, the trivial constant equilibria in the model is obtained in a priori way, that is, any bounded solution (u,v) satisfies (u,v)(0,0) in L(Ω) exponentially, regardless of the size of μ>0.



中文翻译:

具有一般逻辑源和信号产生的完全抛物线拟线性趋化模型的渐近稳定性

在本文中,我们研究了完全抛物趋化系统的整体解的渐近行为: üŤ=düü-小号üv+[Rü-μü1个+σvŤ=Δv-v+üγ,且服从有界和光滑域中的齐次Neumann边界条件 Ω[Rññ2),其中参数 μσγ>0[R[R,以及非线性 d小号C2[0 应该概括原型dü一种0ü+1个-α0小号üb0üü+1个β-1个一种0b0>0αβ[R。我们首先考虑[R>0 并在下提供有界结果 α+β+γ<2ñ, 要么 β+γ<1个+σ, 要么 β+γ=1个+σμ>0。当逻辑源的阻尼效应足够强时,主要结果与渐近稳定性有关。具体来说,有μ0>0 与初始数据无关,因此有界经典解 üv 满足 üv[Rμ1个σ[Rμγσ大号Ω 在以下条件下呈指数增长 μ>μ0[R>0。对于[R<0,模型中的平凡常数均衡是通过先验获得的,即任何有界解üv 满足 üv00大号Ω 指数大小,无论大小 μ>0

更新日期:2019-11-26
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