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Blow-up in second and fourth order semi-linear parabolic partial differentialequations (PDEs) is considered in bounded regions of one, two and three spatialdimensions with uniform initial data. A phenomenon whereby singularities format multiple points simultaneously is exhibited and explained by means of asingular perturbation theory. In the second order case we pre- dict that pointsfurthest from the boundary are selected by the dynamics of the PDE forsingularity. In the fourth order case, singularities can form simultaneously atmultiple locations, even in one spatial dimension. In two spatial dimensions,the singular perturbation theory reveals that the set of possible singularitypoints depends subtly on the geometry of the domain and the equationparameters. In three spatial dimensions, preliminary numerical simulationsindicate that the multiplicity of singularities can be even more complex. Forthe aforementioned scenarios, the analysis highlights the dichotomy ofbehaviors exhibited between the second and fourth order cases.

An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations