Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

Abstract

<p style='text-indent:20px;'>This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t>0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t>0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} (*)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^N(N\geq1) $\end{document}</tex-math></inline-formula> under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> are positive. Based on an <b>new</b> energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mu>\left\{ \begin{array}{ll} {0, \; \; \; {\rm{if}}\; \; N\leq4},\\ {\frac{(N-4)_{+}}{N-2}\max\{1,\lambda_{0}\},\; \; \; {\rm{if}}\; \; N\geq5}\\ \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the initial data <inline-formula><tex-math id="M5">\begin{document}$ (u_0,v_0,w_0) $\end{document}</tex-math></inline-formula> are sufficiently regular. Here <inline-formula><tex-math id="M6">\begin{document}$ \lambda_0 $\end{document}</tex-math></inline-formula> is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.</p>