Bounds for the Steady-State Sel'kov Model for Arbitrary p in Any Number of Dimensions

The Sel'kov model is a system of two differential equations which describe various complex biological and chemical systems. In this system there is an exponent p which must be allowed to be an arbitrary number greater than one according to the underlying model but is usually restricted in mathematical analyses. We show that the restriction is not necessary by proving some a priori estimates for all solutions of the system. The techniques for proof include some maximum principle arguments and the weak Harnack inequality. In fact our techniques apply to a much more general class of problems, including the Brusselator model.