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This work studies front formation in the Allen-Cahn equation with a parameterheterogeneity which slowly varies in space. In particular, we consider aheterogeneity which mediates the local stability of the zero state andsubsequent pitchfork bifurcation to a non-trivial state. For slowly-varyingramps which are either rigidly propagating in time or stationary, we rigorouslyestablish existence and stability of positive, monotone fronts and give leadingorder expansions for their interface location. For non-zero ramp speeds, andsufficiently small ramp slopes, the front location is determined by the localtransition between convective and absolute instability of the base state andleads to an O(1) delay beyond the instantaneous pitchfork location before thesystem jumps to a nontrivial state. The slow ramp induces a further delay ofthe interface controlled by a slow-passage through a fold of strong- andweak-stable eigenspaces of the associated linearization. We introduceprojective coordinates to de-singularize the dynamics near the trivial stateand track relevant invariant manifolds all the way to the fold point. We thenuse geometric singular perturbation theory and blow-up techniques to locate thedesired intersection of invariant manifolds. For stationary ramps, the front isgoverned by the slow passage through the instantaneous pitchfork bifurcationwith inner expansion given by the unique Hastings-McLeod connecting solution ofPainlev\'{e}'s second equation. We once again use geometric singularperturbation theory and blow-up to track invariant manifolds into aneighborhood of the non-hyperbolic point where the ramp passes through zero andto locate intersections.

Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcations