Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production

Abstract

<abstract><p>In this paper, we consider the quasilinear parabolic-elliptic-elliptic attraction-repulsion system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \left\{ \begin{split} &amp;u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w),&amp;\qquad &amp;x\in\Omega,\,t&gt;0, \\ &amp; 0 = \Delta v-\mu_{1}(t)+f_{1}(u),&amp;\qquad &amp;x\in\Omega,\,t&gt;0, \\ &amp;0 = \Delta w-\mu_{2}(t)+f_{2}(u),&amp;\qquad &amp;x\in\Omega,\,t&gt;0 \end{split} \right. \end{equation} $\end{document} </tex-math></disp-formula></p> <p>under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset\mathbb{R}^n, \ n\geq2 $. The nonlinear diffusivity $ D $ and nonlinear signal productions $ f_{1}, f_{2} $ are supposed to extend the prototypes</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation} \nonumber D(s) = (1+s)^{m-1},\ f_{1}(s) = (1+s)^{\gamma_{1}},\ f_{2}(s) = (1+s)^{\gamma_{2}},\ s\geq0,\gamma_{1},\gamma_{2}&gt;0,m\in\mathbb{R}. \end{equation} $\end{document} </tex-math></disp-formula></p> <p>We proved that if $ \gamma_{1} &gt; \gamma_{2} $ and $ 1+\gamma_{1}-m &gt; \frac{2}{n} $, then the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time. However, the system admits a global bounded classical solution for suitable smooth initial datum when $ \gamma_{2} &lt; 1+\gamma_{1} &lt; \frac{2}{n}+m $.</p></abstract>