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I am looking for some references (text books, elementary review papers, journal articles etc) regarding the phenomenon of breakdown in stability for (nonlinear) partial differential equations, i.e if we start off with a partial differential equation and we have a steady state solution and suppose we perturb it, how do we analyse if this leads to stability or instability, I suppose we use semigroup methods and spectral theory amongst other things. What are the various tools, techniques etc that are available. When does linear stability imply nonlinear stability? When does breakdown in stability occur? Are there ways/ theorems to that establish necessary / sufficient conditions for nonlinear instability?

Thank you.

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Look into global attractors. When I took a course on infinite-dimensional dynamical systems and attractor theory, we used the book Infinite-Dimensional Dynamical Systems by Robinson. This is the only book I have ever looked at on the subject so I am sure other users can offer many other recommendations. Google also gives plenty of survey articles full of references. – Henry T. Horton Aug 12 '12 at 20:20
Thank you so much for the reference. Could you please tell me what search string you used and/ or references. I have tried googling to not much effect. – Shibi Vasudevan Aug 12 '12 at 20:28
up vote 1 down vote accepted

If you have access, look at the paper

Optimal gap condition for invariant manifolds, Continuous and Discrete Dynamical Systems, 5 (1999) 233-268

By Latushkin and Layton.

Many of the results are based on methods presented in the book by Daletskii and Krein. This is a book for ODE-s, but in many parts it can be extended nicely to PDEs. And, you can understand the ideas before going into the technicalities.

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Thank you so much for the suggestions. – Shibi Vasudevan Aug 12 '12 at 20:48

There is a book by Temam on infinite dimensional dynamical systems, which is a good reference on such topics.

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Thanks for the reference. – Shibi Vasudevan Aug 13 '12 at 2:19

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