Assess students' ability to analyze stability and local bifurcations of nonlinear evolution PDEs by applying semigroup theory for linearization, spectral and Fredholm-type conditions for operators, and finite-dimensional reductions via center manifold or Lyapunov–Schmidt techniques; tasks include deriving reduced amplitude equations, classifying bifurcation types (saddle-node, pitchfork, Hopf), and determining stability of steady or periodic solutions under appropriate functional-analytic hypotheses (sectoriality, spectral gap, transversality).