Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 6A / Solution

Put $u=\epsilon v+O({\epsilon }^2)$. The linearization at the zero solution isThe appropriate Fourier sine series isOrthogonality of the modes givesThus each mode grows or decays exponentially with its own eigenvalue, and the largest growth rate is the $n=1$ value. Strict linear stability analysis, meaning decay of every mode, is equivalent toAt equality the first mode is neutral in the linearization; the nonlinear terms decide its subsequent behavior. Above this length the first mode grows, giving linear instability.