<p>We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_202_Article_Equa.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="404" /> </MediaObject> <EquationSource Format="TEX">\(\partial_t u = \Delta u - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -\Delta \mathcal{K}_u = u \quad \text{in}\;\; \mathbb{R}^d,\; d = 3,4,\)</EquationSource> </Equation> and derive the final blowup profile <Equation ID="Equb"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_202_Article_Equb.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="304" /> </MediaObject> <EquationSource Format="TEX">\(u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \text{as}\;\; r \to 0, \;\; c_d &gt; 0.\)</EquationSource> </Equation> To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner et al. in [Brenner, Nonlinearity <b>12</b>, 1999].</p>

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Construction of Type I-Log Blowup for the Keller-Segel System in Dimensions 3 and 4

  • Van Tien Nguyen,
  • Nejla Nouaili,
  • Hatem Zaag

摘要

We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system \(\partial_t u = \Delta u - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -\Delta \mathcal{K}_u = u \quad \text{in}\;\; \mathbb{R}^d,\; d = 3,4,\) and derive the final blowup profile \(u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \text{as}\;\; r \to 0, \;\; c_d > 0.\) To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner et al. in [Brenner, Nonlinearity 12, 1999].