Blow-up profile for solutions of a fourth order nonlinear equation

It is well known that the nontrivial solutions of the equation u⁗(r)+κu″(r)+f(u(r))=0u⁗(r)+κu″(r)+f(u(r))=0 blow up in finite time under suitable hypotheses on the initial data, κκ and ff. These solutions blow up with large oscillations. Knowledge of the blow-up profile of these solutions is of great importance, for instance, in studying the dynamics of suspension bridges. The equation is also commonly referred to as extended Fisher–Kolmogorov equation or Swift–Hohenberg equation. In this paper we provide details of the blow-up profile. The key idea is to relate this blow-up profile to the existence of periodic solutions for an auxiliary equation