  We consider a fast-slow partial differential equation (PDE) with
reaction-diffusion dynamics in the fast variable and the slow variable driven
by a differential operator on a bounded domain. Assuming a transcritical normal
form for the reaction term and viewing the slow variable as a dynamic
bifurcation parameter, we analyze the passage through the fast subsystem
bifurcation point. In particular, we employ a spectral Galerkin approximation
and characterize the invariant manifolds for the finite-dimensional Galerkin
approximation for each finite truncation using geometric desingularization via
a blow-up analysis. In addition to the crucial approximation procedure, we also
make the domain dynamic during the blow-up analysis. Finally, we elaborate in
which sense our results approximate the infinite-dimensional problem. Within
our analysis, we find that the PDEs appearing in entry and exit blow-up charts
are quasi-linear free boundary value problems, while in the central/scaling
chart we obtain a PDE, which is often encountered in classical
reaction-diffusion problems exhibiting solutions with finite-time
singularities. In summary, we establish a geometric blow-up analysis for
fast-slow PDEs with a non-hyperbolic point due to a transcritical bifurcation.
Our methodological approach has the potential to deal with the loss of
hyperbolicity for a variety of infinite-dimensional dynamical systems.
