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

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype\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}in a smoothly bounded domain \begin{document}$ \Omega\subset\mathbb{R}^N(N\geq1) $\end{document} under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters \begin{document}$ \mu $\end{document} as well as \begin{document}$ \delta $\end{document} and \begin{document}$ \tau $\end{document} are positive. Based on an new energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever\begin{document}$ \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. $\end{document}and the initial data \begin{document}$ (u_0,v_0,w_0) $\end{document} are sufficiently regular. Here \begin{document}$ \lambda_0 $\end{document} is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.