Boundedness of radial solutions to the parabolic-parabolic Keller--Segel system on the plane with subcritical mass
Abstract
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 8π. 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.
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BibTeX
@misc{hosono2026boundedness,
title = {{Boundedness of radial solutions to the parabolic-parabolic Keller--Segel system on the plane with subcritical mass}},
author = {Tatsuya Hosono},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28230},
eprint = {2610.28230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28230},
}
RIS
TY - UNPB
TI - Boundedness of radial solutions to the parabolic-parabolic Keller--Segel system on the plane with subcritical mass
AU - Tatsuya Hosono
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28230
UR - https://arxiv.org/abs/2610.28230
AB - 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 8π. 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.
ER -