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Abstract. We consider a special case of the Patlak-Keller-Segel system in a disc, which arises in the modelling of chemotaxis phenomena. For a critical value of the total mass, the solutions are known to be global in time but with density becoming unbounded, leading to a phenomenon of mass-concentration in infinite time. We establish the precise grow-up rate and obtain refined asymptotic estimates of the solutions. Unlike in most of the similar, recently studied, grow-up problems, the rate is neither polynomial nor exponential. In fact, the maximum of the density behaves like e √ 2t for large time. In particular, our study provides a rigorous proof of a behaviour suggested by Sire and Chavanis [Phys. Rev. E, 2002] on the basis of formal arguments.

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Abstract.We study the electrostatic MEMS-device equation, ut − ∆u = λ|x| β (1−u) p , with Dirichlet boundary condition. First, we describe the touchdown of non-stationary solution in accordance with the total set of stationary solutions. Then, we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we show the Morse-Smale property for radially symmetric non-stationary solutions. Mathematics Subject Classification (2000). Primary 35K55, 35J60; Secondary 74H35, 74G55, 74K15.

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