Boundedness of the Higher-Dimensional Quasilinear Chemotaxis System with Generalized Logistic Source

Abstract

This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions

$$\begin{cases}u_t=\triangledown\cdot(D(u)\triangledown{u})-\triangledown\cdot(S(u)\triangledown{v})+f(u), & x \in \Omega, t > 0\\v_t=\Delta{v}+w-v, & x \in \Omega, t > 0,\\w_t=u-w, & x \in \Omega, t > 0,\end{cases}$$

in a bounded domain Ω ⊂ Rⁿ(n ≥ 2) with smooth boundary ∂Ω, where the diffusion coefficient D(u) and the chemotactic sensitivity function S(u) are supposed to satisfy D(u) ≥ M₁(u + 1)^−α and S(u) ≤ M₂(u + 1)^β, respectively, where M₁, M₂ > 0 and α, β ∈ R. Moreover, the logistic source f(u) is supposed to satisfy f(u) ≤ a — µu^γ with µ > 0, γ ≥ 1, and a ≥ 0. As $\alpha+2\beta<\gamma-1+\frac{2\gamma}{n}$, we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.