On a Parabolic-ODE system of chemotaxis

Abstract

In this article we consider a coupled system of differential equations to describe the evolution of a biological species. The system consists of two equations, a second order parabolic PDE of nonlinear type coupled to an ODE. The system contains chemotactic terms with constant chemotaxis coefficient describing the evolution of a biological species \begin{document}$u$\end{document} which moves towards a higher concentration of a chemical species \begin{document}$v$\end{document} in a bounded domain of \begin{document}$ \mathbb{R}^n$\end{document} . The chemical \begin{document}$v$\end{document} is assumed to be a non-diffusive substance or with neglectable diffusion properties, satisfying the equation \begin{document}$v_t = h(u, v).$ \end{document} We obtain results concerning the bifurcation of constant steady states under the assumption \begin{document} $ h_v+χ u h_u>0 $ \end{document} with growth terms \begin{document}$g$\end{document} . The Parabolic-ODE problem is also considered for the case \begin{document}$h_v+χ u h_u = 0$\end{document} without growth terms, i.e. \begin{document}$g \equiv 0$\end{document} . Global existence of solutions is obtained for a range of initial data.