Boundedness and blow up for the general activator-inhibitor model

Abstract

This paper deals with the general Activator-inhibitor model

$$\{ \begin{array}{*{20}c} {u_t = d\Delta u - \mu u + u^p v^{ - q} + \sigma ,} \\ {v_t = D\Delta v - \nu u + u^r v^{ - s} } \\\end{array}$$

with Neumann boundary conditions. We show that the solutions of the model are bounded all the time for each pair of initial values ifr>p−1 andrq>(p−1)(s−1), and that they will blow up in a finite time for some initial values if eitherr>p−1 withrq<(p−1)(s+1) orr<p−1.