Finite-Time Blowup in a Supercritical Quasilinear Parabolic-Parabolic Keller-Segel System in Dimension 2

Abstract

In this note we show finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. We prove this result by slightly adapting M. Winkler's method, which he introduced in (Winkler in J. Math. Pures Appl., 10.1016/j.matpur.2013.01.020, 2013) for the semilinear Keller-Segel system in dimensions at least three, to the two-dimensional setting. This is done in the case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned non-decay assumption is essential with respect to keeping the finitetime blowup result. Namely, we prove that without the non-decay assumption solutions exist globally in time, however infinite-time blowup may occur.

Boundedness in a fully parabolic chemotaxis system with singular sensitivity

“…Indeed, in this context, Kavallaris and Souplet [10] studied a precise grow-up rate and asymptotic estimates for solutions to a simplified chemotaxis system without 1 v . Moreover, as to the problem (1.1) without 1 v , Cieślak and Stinner [3] showed that the solutions blow up in finite time under some conditions. As to the present problem (1.1) with 1 v , global existence of weak solutions was established when χ < n+2 3n−4 ( [16]).…”

Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with a logistic source

“…During the past decades, the Keller-Segel models of type (1.1) have been studied extensively by many authors, where the main issue of the investigation is whether the solutions of the models are bounded or blow-up (see e.g., Cieślak et al [7,4,5,6], Calvez and Carrillo [2], Keller and Segel [15,16], Horstmann et al [11,12,13], Osaki [20,19], Painter and Hillen [21], Perthame [22], Rascle and Ziti [23], Wang et al [28,29], Winkler [31,32,33,34,35,37], Zheng [39]). Especially, in the absence of the logistic source (i.e.…”

Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring

“…In a fully parabolic system, though the global existence results for initial data with mass less than m * are known, finite-time blowup is still an open question. The only results on finite-time blowup available in a fully parabolic case are those in [7] where there has been constructed some special radially symmetric initial data yielding a finite-time singularity and recent proof of finite-time blowup in a fully parabolic case, see [6], however only in a quasilinear supercritical model. In dimensions n ≥ 3 it has been known for a long time that radially symmetric solutions to the parabolic-elliptic simplification blow up in finite time independently of the magnitude of initial mass, see [10].…”