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Effects of boundary conditions on spatially periodic states. (English) Zbl 0599.76099
We investigate analytically and numerically the influence of linear homogeneous boundary conditions on the stationary solutions of a simple model for cellular pattern formation in one dimension. For all boundary conditions there exists in a reduced wavenumber band at least one static solution where the amplitude falls below its bulk value near the boundary (type-I solution). A linear stability analysis of the uniform state at threshold reveals that type-I solutions are often unstable. Then there exists in the full-Eckhaus-stable band, a static solution where the amplitude rises above its bulk value near the boundary (”type-II” solution), or a limit-cycle solution where the amplitude near the boundary oscillates. These solutions bifurcate from the homogeneous state below the bulk threshold and therefore remain finite at threshold.

MSC:
76Rxx Diffusion and convection
35B32 Bifurcations in context of PDEs
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