Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme. (English) Zbl 1323.35087
The authors study a reaction-diffusion system with Degn-Harrison reaction scheme and with Neumann boundary value condition. The domain here is \(\Omega\subset\mathbb{R}^n\). Assuming that \((u(x),v(x))\) is any positive nonconstant steady state, some basic properties of \((u(x),v(x))\) are given. They then discuss the stability of a constant steady state, both for the ODE and the PDE model. Their result indicates that if either the size of the reactor or the effective diffusion rate is large, then there exists no positive nonconstant steady state. Finally, assuming that \(\Omega=(0,\pi)\), the structure of a positive nonconstant steady state is investigated: the local structure of the steady state bifurcation from double eigenvalues by using the techniques of space decomposition and the implicit theorem, and the global structure of the steady state bifurcation from simple eigenvalues by using the bifurcation theorem.
Reviewer: Peixuan Weng (Guangzhou)
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B35 | Stability in context of PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
Keywords:
Degn-Harrison; fundamental properties; stability; nonconstant positive steady state; local and nonlocal structureReferences:
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