THE NON-EXISTENCE AND EXISTENCE OF NON-CONSTANT POSITIVE SOLUTIONS FOR A DIFFUSIVE AUTOCATALYSIS MODEL WITH SATURATION

This paper deals with a diffusive autocatalysis model with saturation under Neumann boundary conditions. Firstly, some stability and Turing instability results are obtained. Then by the maximum principle, H$ \ddot{o} $lder inequality and Poincar$ \acute{e} $ inequality, a priori estimates and some basic characterizations of non-constant positive solutions are given. Moreover, some non-existence results are presented for three different situations. In particular, we find that the model does not have any non-constant positive solution when the parameter which represents the saturation rate is large enough. In addition, we use the theories of Leray-Schauder degree and bifurcation to get the existence of non-constant positive solutions, respectively. The steady-state bifurcations at both simple and double eigenvalues are intensively studied and we establish some specific condition to determine the bifurcation direction. Finally, a few of numerical simulations are provided to illustrate theoretical results.