@article{Ferreira2022, 
author = {Lucas C. F. Ferreira},
title = {On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical        L          p      -space},
year = {2022},
journal = {Mathematics in Engineering},
volume = {4},
number = {6},
pages = {1-14},
keywords = {Keller-Segel system, uniqueness, critical spaces, bilinear estimates, Lorentz spaces},
url = {https://www.sciopen.com/article/10.3934/mine.2022048},
doi = {10.3934/mine.2022048},
abstract = {We are concerned with the uniqueness of mild solutions in the critical Lebesgue space        L                  n        2              (            R              n        ) for the parabolic-elliptic Keller-Segel system,    n  ≥  4. For that, we prove the bicontinuity of the bilinear term of the mild formulation in the critical weak-       L                  n        2             space, without using Kato time-weighted norms, time-spatial mixed Lebesgue norms (i.e.,        L          q        (  (  0  ,  T  )  ;      L          p        )-norms with    q  ≠  ∞), and any other auxiliary norms. Our proofs are based on Yamazaki's estimate, duality and Hölder's inequality, as well as an adapted Meyer-type argument. Since they are different from those of Kozono, Sugiyama and Yahagi [J. Diff. Eq. 253 (2012)] and it is not clear whether mild solutions are weak solutions in the critical    C  (  [  0  ,  T  )  ;      L                  n        2              ), our results complement theirs in a twofold way. Moreover, the bilinear estimate together heat semigroup estimates yield a well-posedness result whose dependence with respect to the decay rate    γ of the chemoattractant is also analyzed.}
}