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Bifurcation for a reaction-diffusion system with unilateral and Neumann boundary conditions. (English) Zbl 1237.35013
The authors study bifurcations of stationary solutions of the reaction-diffusion system \[ \frac{du}{dt}=d_1\Delta u+b_{11}u+b_{12}v+f_1(u,v),\;\frac{dv}{dt}=d_2\Delta v+b_{21}u+b_{22}v+f_2(u,v) \] in a bounded domain \(\Omega\subseteq \mathbb{R}^N\) with Neumann-Signorini boundary conditions \[ \frac{\partial u}{\partial n}=0\;\text{on}\;\partial \Omega;\;v\geq 0,\;\frac{\partial v}{\partial n}\geq 0,\;\frac{\partial v}{\partial n} v=0\;\text{on}\;\Gamma;\;\frac{\partial v}{\partial n}= 0\;\text{on}\;(\partial \Omega)\setminus \Gamma,\;\Gamma\subseteq \Omega. \] The diffusion coefficients \(d=(d_1,d_2)\in \mathbb{R}_+^2\) are considered as bifurcation parameters and \(f_j\) represent “higher order terms” of some nonlinearity. The assumptions concerning the real coefficients \(b_{ij}\) guarantee that Turing’s effect of diffusion-driven instability with purely Neumann conditions \(\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0\;\text{on}\;\partial \Omega\) occurs.

MSC:
35B32 Bifurcations in context of PDEs
35K57 Reaction-diffusion equations
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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