Construction of type I-Log blowup for the Keller-Segel system in dimensions 3 and 4

2023/09/25 by Nguyen, V. T., Nouaili, N., Zaag, H. · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.13932

Abstract

We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system ∂_t u = Δu - ∇ ⋅ (u ∇ K_u), -ΔK_u = u \textupin ℝ^d, d = 3,4, and derive the final blowup profile u(r,T) ∼ c_d (|log r|^(d-2)/(d))/(r²) \textupas r → 0, c_d > 0. To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999).

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