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?