Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production

By Mengyao Ding, Wei Wang. 2017

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In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: \begin{document}$ u_t = \nabla\cdot(D(u)\nabla u-S(u)\nabla v) $\end{document} , \begin{document}$ \tau_1v_t = \Delta v-a_1v+b_1w $\end{document} , \begin{document}$ \tau_2w_t = \Delta w-a_2w+b_2u $\end{document} , under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset\Bbb{R}^{n} $\end{document} ( \begin{document}$ n\geq 1 $\end{document} ), where \begin{document}$ \tau_i,a_i,b_i>0 $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) are constants, and the diffusivity \begin{document}$ D $\end{document} and the density-dependent sensitivity \begin{document}$ S $\end{document} satisfy \begin{document}$ D(s)\geq a_0(s+1)^{-\alpha} $\end{document} and \begin{document}$ 0\leq S(s)\leq b_0(s+1)^{\beta} $\end{document} for all \begin{document}$ s\geq 0 $\end{document} with \begin{document}$ a_0,b_0>0 $\end{document} and \begin{document}$ \alpha,\beta\in\Bbb R $\end{document} . It is proved that if \begin{document}$ \alpha+\beta and \begin{document}$ n = 1 $\end{document} , or \begin{document}$ \alpha+\beta with \begin{document}$ n\geq 2 $\end{document} , for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: \begin{document}$ u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla z)+\xi\nabla\cdot(u\nabla w) $\end{document} , \begin{document}$ z_t = \Delta z-\rho z+\mu u $\end{document} , \begin{document}$ w_t = \Delta w-\delta w+\gamma u $\end{document} , where \begin{document}$ \chi,\mu,\xi,\gamma,\rho,\delta>0 $\end{document} , and the diffusivity \begin{document}$ D $\end{document} fulfills \begin{document}$ D(s)\geq c_0(s+1)^{M-1} $\end{document} for any \begin{document}$ s\geq 0 $\end{document} with \begin{document}$ c_0>0 $\end{document} and \begin{document}$ M\in\Bbb R $\end{document} . As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. \begin{document}$ \chi\mu = \xi\gamma $\end{document} ), the solution is globally bounded if \begin{document}$ M>-1 $\end{document} and \begin{document}$ n = 1 $\end{document} , or \begin{document}$ M>2-4/n $\end{document} with \begin{document}$ n\geq 2 $\end{document} . This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/601c44044b0ddcfcd5f6979098986e0277e3c018