The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model

Abstract

<abstract><p>We study the following quasilinear pursuit-evasion model:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi\nabla \cdot (u(u+1)^{\alpha}\nabla w)+u(\lambda_{1}-\mu_{1}u^{r_{1}-1}+ av),\ &amp;\ \ x\in \Omega, \ t&gt;0,\\[2.5mm] v_{t} = \Delta v+\xi\nabla \cdot(v(v+1)^{\beta}\nabla z)+v(\lambda_{2}-\mu_{2}v^{r_{2}-1}-bu), \ &amp;\ \ x\in \Omega, \ t&gt;0,\\[2.5mm] 0 = \Delta w-w+v, \ &amp;\ \ x\in \Omega, \ t&gt;0 ,\\[2.5mm] 0 = \Delta z-z+u,\ &amp;\ \ x\in \Omega, \ t&gt;0 , \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1), $ where $ a, b, \chi, \xi, \lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2} &gt; 0, $ $ \alpha, \beta \in\mathbb{R}, $ and $ r_{1}, r_{2} &gt; 1. $ When $ r_{1} &gt; \max\{1, 1+\alpha\}, r_{2} &gt; \max\{1, 1+\beta\}, $ it has been proved that if $ \min\{(r_{1}-1)(r_{2}-\beta-1), (r_{1}-\alpha-1)(r_{2}-\beta-1)\} &gt; \frac{(n-2)_{+}}{n}, $ then for some suitable nonnegative initial data $ u_{0} $ and $ v_{0}, $ the system admits a unique globally classical solution which is bounded in $ \Omega\times(0, \infty) $.</p></abstract>