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Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals

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  • We consider the chemotaxis-growth system

    $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \chi \nabla \cdot (u\nabla v) + \mu u(1 - u),}&{x \in \Omega ,{\mkern 1mu} t > 0,}\\ {{v_t} = \Delta v - v + h(u),}&{x \in \Omega ,{\mkern 1mu} t > 0,} \end{array}} \right.$

    under no-flux boundary conditions, in a convex bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary, where $χ>0$ and $μ>0$ are given parameters, and $h(s)$ is a prescribed function on $[0, ∞)$ .

    It is shown that under the assumption that

    $4|{h}'|<\sqrt{2\mu -7{{\chi }^{2}}},$

    for any given nonnegative $u_0∈ C^0(\bar{Ω})$ and $v_0∈ W^{1, ∞}(Ω)$ the system possesses a global classical solution which is bounded in $Ω× (0, ∞)$ . Moreover, whenever

    $χ |h'| < \sqrt{8μ},$

    any bounded classical solution constructed above stabilizes to the constant stationary solution $(1, h(1))$ as the time goes to infinity.

    Mathematics Subject Classification: Primary:35B40, 35K57, 35Q92, 92C17.

    Citation:

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