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Boundedness in prey-taxis system with rotational flux terms

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The authors are supported in part by the National Natural Science Foundation of China (No.11671079, 11701290, 11601127 and 11171063), the Natural Science Foundation of Jiangsu Province(No.BK20170896), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No.KYCX19_0054)

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  • This paper investigates prey-taxis system with rotational flux terms

    $ \begin{equation*} \begin{cases} u_t = \Delta u-\nabla\cdot(uS(x,u,v)\nabla v)+uv-\rho u,& x\in\Omega,\ t>0,\\ v_t = \Delta v-\xi uv+\mu v(1-\alpha v),&x\in\Omega,\ t>0, \end{cases} \end{equation*} $

    under no-flux boundary conditions in a bounded domain $ \Omega\subset\mathbb{R}^n\; (n\geq1) $ with smooth boundary. Here the matrix-valued function $ S\in C^2(\bar{\Omega}\times[0,\infty)^2;\mathbb{R}^{n\times n}) $ fulfills $ |S(x,u,v)|\leq\frac{S_0(v)}{(1+u)^\theta}(\theta\geq0) $ for all $ (x,u,v)\in\bar{\Omega}\times[0,\infty)^2 $ with some nondecreasing function $ S_0 $. It is proved that for nonnegative initial data $ u_0\in C^0(\overline{\Omega}) $ and $ v_0\in W^{1,q}(\Omega) $ with some $ q>\max\{n,2\} $, if one of the following assumptions holds: (i) $ n = 1 $, (ii) $ n\geq2, \theta = 0 $ and $ S_0(m)m<\frac{2}{\sqrt{3n(11n+2)}} $, (iii) $ \theta>0 $, then the model possesses a global classical solution that is uniformly bounded. Where $ m: = \max\{\|v_0\|_{L^\infty(\Omega)}, \frac{1}{\alpha}\} $.

    Mathematics Subject Classification: 35B45, 35K55, 35K65, 92C17.

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