This article proves the ill-posedness of the Cauchy problem for the two-dimensional Keller–Segel model in Triebel–Lizorkin spaces, for . In particular, it is shown that solutions can develop norm inflation under certain settings in that the solution can become arbitrarily large after an arbitrarily short time even for small initial data.
Deng, Chao and Villavert, John, "Ill-posedness of the two-dimensional Keller-Segel model in Triebel-Lizorkin spaces" (2014). Mathematical and Statistical Sciences Faculty Publications and Presentations. 52.
Nonlinear Analysis: Theory, Methods & Applications