Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ2 with Critical Mass

Abstract

We consider the Cauchy problem for a parabolic-elliptic system in ℝ2{\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 is 8⁢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.

Keywords

• parabolic-elliptic system
• keller–segel equation
• chemotaxis system
• critical mass, boundedness of solutions
• 35b45
• 35k15
• 35k55