Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

Abstract

<p style='text-indent:20px;'>This paper deals with the classical solution of the following chemotaxis system with generalized logistic growth and indirect signal production</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1"> \begin{document}$ \begin{eqnarray} \left\{ \begin{array}{llll} u_t = \epsilon\Delta u-\nabla\cdot(u\nabla v)+ru-\mu u^\theta, &\\ 0 = d_1\Delta v-\beta v+\alpha w, &\\ 0 = d_2\Delta w-\delta w+\gamma u, & \end{array} \right. \end{eqnarray} \quad\quad\quad\quad(1)$ \end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>and the so-called strong <inline-formula><tex-math id="M1">\begin{document}$ W^{1, q}( \Omega) $\end{document}</tex-math></inline-formula>-solution of hyperbolic-elliptic-elliptic model</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE2"> \begin{document}$ \begin{eqnarray} \left\{ \begin{array}{llll} u_t = -\nabla\cdot(u\nabla v)+ru-\mu u^\theta, &\\ 0 = d_1\Delta v-\beta v+\alpha w, &\\ 0 = d_2\Delta w-\delta w+\gamma u, & \end{array} \right. \end{eqnarray} \quad\quad\quad\quad(2)$ \end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>in arbitrary bounded domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ r, \mu, d_1, d_2, \alpha, \beta, \gamma, \delta>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \theta>1 $\end{document}</tex-math></inline-formula>. Via applying the viscosity vanishing method, we first prove that the classical solution of (1) will converge to the strong <inline-formula><tex-math id="M6">\begin{document}$ W^{1, q}( \Omega) $\end{document}</tex-math></inline-formula>-solution of (2) as <inline-formula><tex-math id="M7">\begin{document}$ \epsilon\rightarrow0 $\end{document}</tex-math></inline-formula>. After structuring the local well-pose of (2), we find that the strong <inline-formula><tex-math id="M8">\begin{document}$ W^{1, q}( \Omega) $\end{document}</tex-math></inline-formula>-solution will blow up in finite time with non-radial symmetry setting if <inline-formula><tex-math id="M9">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded convex domain, <inline-formula><tex-math id="M10">\begin{document}$ \theta\in(1, 2] $\end{document}</tex-math></inline-formula>, and the initial data is suitable large. Moreover, for any positive constant <inline-formula><tex-math id="M11">\begin{document}$ M $\end{document}</tex-math></inline-formula> and the classical solution of (1), if we add another hypothesis that there exists positive constant <inline-formula><tex-math id="M12">\begin{document}$ \epsilon_0(M) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \epsilon\in(0,\ \epsilon_0(M)) $\end{document}</tex-math></inline-formula>, then the classical solution of (1) can exceed arbitrarily large finite value in the sense: one can find some points <inline-formula><tex-math id="M14">\begin{document}$ \left(\tilde{x}, \tilde{t}\right) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M15">\begin{document}$ u(\tilde{x}, \tilde{t})>M $\end{document}</tex-math></inline-formula>.</p>