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Infinite time aggregation for the critical Patlak-Keller-Segel model in \(\mathbb R^2\). (English) Zbl 1155.35100
For the two-dimensional Smoluchowski-Poisson equation (also called the parabolic-elliptic Keller-Segel chemotaxis model)
\[ \begin{aligned} \partial_t n &= \nabla\cdot (\nabla n-\chi n \nabla c), \quad (t,x)\in (0,\infty)\times{\mathbb R}^2,\\ c(t,x)&=-\frac{1}{2\pi} \int_{{\mathbb R}^2} \ln| x-y| n(t,y)\,dy, \quad (t,x)\in (0,\infty)\times{\mathbb R}^2,\\ n(0,x)& = n_0(x)\geq 0, \quad x\in {\mathbb R}^2, \end{aligned} \] the value of the \(L^1\)-norm of \(n_0\) governs the maximal existence time of the solution \(n\). More precisely, given \(\chi>0\), if \(\chi \| n_0\| _1>8\pi\), then the solution becomes unbounded in finite time while the solution is global and bounded if \(\chi \| n_0\| _1<8\pi\). The critical case \(\chi \| n_0\| _1=8\pi\) is analysed in the paper under review: the solution is proved to be global and becomes unbounded as \(t\to\infty\). More precisely, \(n(t)\) behaves as \((8\pi/\chi)\delta_{x_M}\) as \(t\to\infty\), \(x_M\) denoting the centre of mass of \(n_0\) and \(\delta_{x_M}\) the Dirac measure centred at \(x_M\).

MSC:
35Q80 Applications of PDE in areas other than physics (MSC2000)
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
92C17 Cell movement (chemotaxis, etc.)
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