Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation

We consider the slow passage through a Turing bifurcation in the Swift-Hohenberg equation. We generalise the formally derived multiple scales ansatz from modulation theory for use in the slowly time-dependent setting. The key technique is to reformulate the problem via a geometric blow-up transformation. This leads to non-autonomous modulation equations of Ginzburg-Landau type in the blown-up space. We analyse solutions to the modulation equations in weighted Sobolev spaces in two different cases: (i) A symmetric case featuring delayed stability loss, and (ii) A non-symmetric case with a source term. Rigorous estimates on the error of the dynamic modulation approximation are derived in order to characterise the dynamics of the Swift-Hohenberg equation. This allows for a detailed asymptotic description of solutions to the original Swift-Hohenberg equation in both cases (i)-(ii). We also prove the existence of delayed stability loss in case (i), and provide a lower bound for the delay time.