A blow-up result for the chemotaxis system with nonlinear signal production and logistic source

Abstract

<p style='text-indent:20px;'>In this paper we consider the following chemotaxis-growth system with nonlinear signal production and logistic source <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi\nabla\cdot( u\nabla v)+\lambda u-\mu u^{\alpha}, \quad &amp;x\in \Omega, t&gt;0, \\ 0 = \Delta v-\mu (t)+f(u), \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}f(u(\cdot, t)), \quad &amp;x\in \Omega, t&gt;0, \ \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>with homogeneous Neumann boundary conditions in the ball <inline-formula><tex-math id="M1">\begin{document}$ \Omega = B_R(0)\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ R&gt;0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ \chi, \lambda, \mu&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \alpha&gt;1 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> is an appropriate regular function satisfying <inline-formula><tex-math id="M7">\begin{document}$ f(u)\geq ku^{\kappa} $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M8">\begin{document}$ u\geq1, \kappa&gt;0 $\end{document}</tex-math></inline-formula> with some constants <inline-formula><tex-math id="M9">\begin{document}$ k&gt;0 $\end{document}</tex-math></inline-formula>. If the number <inline-formula><tex-math id="M10">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> satisfy <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \kappa+1&gt;\alpha\left(\frac{2}{n}+1\right), \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>then there exists appropriate initial data such that the corresponding solution <inline-formula><tex-math id="M12">\begin{document}$ (u, v) $\end{document}</tex-math></inline-formula> of the system blow up in finite time. This result extends the blow-up result of the chemotaxis model without logistic cell kinetics in [<xref ref-type="bibr" rid="b45">45</xref>]. Apparently, for the case <inline-formula><tex-math id="M13">\begin{document}$ \kappa = 1 $\end{document}</tex-math></inline-formula>, this provides a rigorous detection for blow-up of solution in spaces-dimensions three and four.

Topics & Concepts