Steady State Solution Of The Generalized Gray-scott Model

The generalized Gray-Scott model can be used to describe the reaction diffusion phenomenon between chemical reactants. Mathematically, the model is represented by reaction diffusion equations, whose steady state solutions satisfy the following elliptic equations: where ? is RN a bounded domain with a smooth boundary, v is the outward unit normal vector on (?)? and (?)v=(?)/(?)v. The parameters d1,d2,F,k,p are always assumed to be positive constants. We mainly study the existence and non-existence of non-constant positive steady state solutions of the problem (0.1). Main results are as follows:1. In the case of p?(0,1], we show that the problem (0.1) does not exist positive solutions. In the case of p>1, the sufficient conditions for the existence of the problem (0.1) are given. In particular, the results on the case of p=2 improve the corresponding results in the literature.2. By using the topological degree theory, the existence of the positive steady state solution is studied.3. For a similar model, namely, the Sel'kov model, it is proved, by a similar approach, that if p?(0,1], then the problem does not admit any positive solution, which greatly improves some previous works.