Boundedness of radial solutions to the parabolic-parabolic Keller--Segel system on the plane with subcritical mass

We establish uniform-in-time boundedness of radially symmetric global solutions to the two-dimensional parabolic-parabolic Keller–Segel system on the whole space with mass below $8\pi$. While global existence is known throughout this subcritical mass range, uniform-in-time boundedness has remained open. On bounded domains, the free energy method of Nagai, Senba and Yoshida (1997) yields both global existence and boundedness under the optimal mass condition. On the whole space, however, the corresponding estimates do not rule out growth as time tends to infinity, preventing a direct application of this approach to establish boundedness. The key ingredients are new interior estimates controlling the localized free energy in terms of its dissipation, together with uniform-in-time exterior estimates exploiting radial symmetry.