Premium
Instability of Spatially — Periodic States for a Family of Semilinear PDE's on an Infinite Strip
Author(s) -
Bridges Thomas J.,
Mielke Alexander
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19961790102
Subject(s) - mathematics , center manifold , instability , bounded function , manifold (fluid mechanics) , invariant manifold , norm (philosophy) , mathematical analysis , center (category theory) , hamiltonian (control theory) , pure mathematics , nonlinear system , bifurcation , law , physics , mathematical optimization , mechanical engineering , chemistry , hopf bifurcation , quantum mechanics , political science , mechanics , engineering , crystallography
A family of time‐evolution equations, for which the time‐independent part is a semilinear elliptic equation on an infinite strip, is considered with attention to the instability of spatially‐periodic states. The analysis uses a sequence of center ‐ manifold reductions; first, every local, small norm, bounded solution is contained in a center manifold. Second, it is proved that every local, small norm, solution of the linear stability problem is also contained in a related non‐autonomous center manifold. For the case where the center manifold is foliated by periodic states, with wavelength L ( s ), parametrized by s , the level sets of a suitably defined spatial Hamiltonian functional, we prove a geometric criterion for instability: a spatially periodic state is linearly unstable in the time—dependent problem if L' ( s ) > 0. Properties of the manifold of periodic states and their linear stability are explicitly constructed for a family of potentials.