Blowup Patterns and Weak Concentration Rates in the Smoluchowski-poisson Equation in Higher Dimensions

We study the Smoluchowski-Poisson equation on the whole space Rn for n≥3. In particular, we introduce the notion of a blowup point with strong pattern and weak concentration rate and prove that the Hausdorff dimension of the set of such points is at mostn−2. Despite the extensive literature on blowup phenomena in nonlinear PDEs, many fundamental questions remain unresolved, especially,regarding the intricate geometry of the blowup set in high-dimensional settings. In this work, we refine existing methods and introduce a novel framework for quantifying the interplay between concentration rates and spatial structure. Our results yield sharper bounds on the Hausdorff dimension, opening new avenues for understanding critical thresholds in models ranging from chemotaxis to gravitational dynamics.