The 8π‐problem for radially symmetric solutions of a chemotaxis model in the plane

Abstract

SUMMARYThe existence, uniqueness and large time behaviour of radially symmetric solutions to a chemotaxis system in the plane R 2 are studied either for the critical value of the mass equal to 8 or in the subcritical case.

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

“…Let us observe that for τ > τ * the function M (a, τ ) depends on a in a nonmonotone manner. This is a significant difference with the monotone dependence of self-similar solutions of the parabolic-elliptic Keller-Segel system (see [3,Sec. 4]).…”

“…However, it has not yet been proved that explosion occurs in finite time as soon as M > 8 π, for instance under some additional assumptions like a smallness condition on R R 2 |x| 2 n 0 (x) dx. If M = 8 π, there is an infinite number of steady states (see [3]), but no other result is available, apart from self-similar solutions.…”

Stability of Stationary Solutions and Existence of Oscillating Solutions to a Chemotaxis System in High Dimensional Spaces

“…However, there exist solutions having more complicated behavior. In two dimensional case, bounded oscillating solutions and unbounded oscillating solutions were found (see [10]), by using a similar argument to [19] and the stability of stationary solutions shown in [2]. In [19], the following Cauchy problem u t ¼ Du þ u p in R n  ð0; yÞ; uðÁ; 0Þ ¼ u 0 in R n was considered, the existence of bounded oscillating solutions and unbounded oscillating solutions were constructed, and in the proof it is important that each stationary solution is stable and that stationary solutions layer, where n b 11 and p b p JL ¼ fðn À 2Þ 2 À 4n þ 8 ffiffiffiffiffiffiffiffiffiffiffi n À 1 p g=fðn À 2Þðn À 10Þg.…”

“…Here, k Á k L p is the standard L p ðR n Þ norm for p A ½1; y . In two dimensional case, it was shown that radial stationary solutions are stable under radial and small perturbations (see [2]), which means that some radial solutions converge to a stationary solution.…”

“…Concerning the proof of stability of stationary solutions, Lyapunov functions play an important role in two dimensional chemotaxis model (see [2]), and the convexity of the nonlinear term plays an important role in the nonlinear Fujita type heat equation (see [19]). However, in high dimensional case, our system has neither of these properties.…”