Boundedness and stabilization in a quasilinear chemotaxis model with nonlocal growth term and indirect signal production

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Abstract

In this paper, we study the following fully parabolic chemotaxis system with nonlocal growth and indirect signal production: $\left\{\begin{array}{ll} u_{t}=\nabla \cdot(D(u) \nabla u)-\nabla \cdot(u \nabla v)+f(u), & x \in \Omega, t>0, \\ v_{t}=\Delta v+w-v, & x \in \Omega, t>0, \\ w_{t}=\Delta w+u-w, & x \in \Omega, t>0, \\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0, & x \in \partial \Omega, t>0, \\ u(x, 0)=u_{0}(x), v(x, 0)=v_{0}(x), w(x, 0)=w_{0}(x), & x \in \Omega, \end{array}\right.$ in a smooth bounded domain $\Omega \subset \mathbb{R}^{n}$(n ≥ 3), where D(u) ≥ D₁ u^γ,$f(u)=u\left(a_{0}-a_{1} u^{\sigma}+a_{2} \int_{\Omega} u^{\sigma} d x\right)$, D₁,a₁,a₂ are positive constants, a₀, $\gamma \in \mathbb{R}$, σ ≥ max{1, −γ} and a₁ − a₂|Ω| > 0. It is shown that the above system admits a globally bounded classical solution if $\frac{4}{n}+\frac{\gamma}{\sigma}>\frac{3-\sigma}{1+\sigma}$. Furthermore, by the method of Lyapunov functionals, the global stability of steady states with convergence rates is established.

Keywords

chemotaxis system / global existence / boundedness / nonlocal term / asymptotic stability

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