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We consider the Cauchy problem for a parabolic-elliptic system inℝ  2{\mathbb{R}^{2}}   , called the parabolic-elliptic Keller–Segel equation, which appears in various fields in biology and physics. In the critical mass case where the total mass of the initial data is8  ⁢  π{8\pi}   , the unboundedness of nonnegative solutions to the Cauchy problem was shown by Blanchet, Carrillo and Masmoudi [ 7 ] under some conditions on the initial data, on the other hand, conditions for boundedness were given by Blanchet, Carlen and Carrillo [ 6 ] and López-Gómez, Nagai and Yamada [ 23 ]. In this paper, we investigate further the boundedness of nonnegative solutions.

Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ 2 with Critical Mass