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Nonlinear diffusive stability of spatially periodic solutions – abstract theorem and higher space dimensions –. (English) Zbl 0907.35015
Nishiura, Y. (ed.) et al., Proceedings of the international conference on asymptotics in nonlinear diffusive systems – towards the understanding of singularities in dissipative structures –, Sendai, Japan, July 28–August 1, 1997. Sendai: Tohoku Univ. Tohoku Math. Publ. 8, 159-167 (1998).
Summary: We consider the nonlinear stability of a spatially periodic equilibrium in a spatially extended domain. The eigenfunctions of the associated linearization are given by Bloch waves. The spectrum is continuous and contains zero provided that the equilibrium is not trivial. If the spectrum lies completely to the left hand side of the imaginary axis the diffusive stability method is used to establish the nonlinear stability of the equilibrium. The perturbations decay to zero similar to a solution of a linear diffusion equation. Here, we give an abstract version of this stability result and concentrate on the case of more than one unbounded space direction.
For the entire collection see [Zbl 0886.00019].

35B35 Stability in context of PDEs
35B10 Periodic solutions to PDEs
37C75 Stability theory for smooth dynamical systems