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On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law. (English) Zbl 1148.35094

Summary: Understanding of spatial and temporal behaviour of interacting species or reactants in ecological or chemical systems has become a central issue, and rigorously determining the formation of patterns in models from various mechanisms is of particular interest to applied mathematicians. In this paper, we study a bimolecular autocatalytic reaction-diffusion model with saturation law and are mainly concerned with the corresponding steady-state problem subject to the homogeneous Neumann boundary condition. In particular, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain reactant is large or small. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system. Our theoretical analysis shows that the diffusion rate of this reactant and the size of the reactor play decisive roles in leading to the formation of stationary patterns.

MSC:

35Q80 Applications of PDE in areas other than physics (MSC2000)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B45 A priori estimates in context of PDEs
92C15 Developmental biology, pattern formation
92C40 Biochemistry, molecular biology
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