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BOUNDEDNESS OF THE HIGHER-DIMENSIONAL QUASILINEAR CHEMOTAXIS SYSTEM WITH GENERALIZED LOGISTIC SOURCE
Qingquan TANG, Qiao XIN, Chunlai MU
Acta mathematica scientia,Series B. 2020, 40 (3):
713-722.
DOI: 10.1007/s10473-020-0309-0
This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions \[ \left\{ \begin{array}{lll} u_t=\nabla\cdot(D(u)\nabla u)-\nabla \cdot (S(u)\nabla v)+f(u), &x\in \Omega,t>0\\ v_t = \Delta v+w-v, &x\in \Omega, t>0,\\ w_t=u-w, &x\in \Omega, t>0, \end{array} \right.\] in a bounded domain $\Omega \subset R^{n}(n\geq 2)$ with smooth boundary $\partial\Omega$, where the diffusion coefficient $D(u)$ and the chemotactic sensitivity function $S(u)$ are supposed to satisfy $D(u)\geq M_{1}(u+1)^{-\alpha}$ and $S(u)\leq M_{2}(u+1)^\beta$, respectively, where $M_{1},M_{2}>0$ and $\alpha, \beta\in R$. Moreover, the logistic source $f(u)$ is supposed to satisfy $f(u)\leq a-\mu u^{\gamma}$ with $\mu>0$, $\gamma\geq 1$, and $a\geq 0$. 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.
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