Finite-time aggregation into a single point in a reaction - diffusion system

Abstract

We consider the following system: (S) u t = u − χ∇(u∇v) χ > 0 v = 1 − u which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dirac mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed.

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

“…Actually, the radial case is better understood and, in two space dimensions for large mass M (larger than the corresponding K 0 in Theorem 2.1), the type of blow-up has been specified. In [32] the authors proved that chemotactic collapse i.e. pointwise concentration as a Dirac mass occurs, more precisely, we have the following Theorem 2.5 (Chemotactic collapse for system (1)).…”

“…It can be derived just dividing this integral in two integrals for |x| ≤ R (and use Hölder inequality) and |x| ≥ R (and use |x| 2 ≥ R 2 ) and optimizing the result in R. In three dimensions, it is an open question to replace assumption (3) by " n 0 L d 2 (R d ) large enough" (without second xmoment), as it is suggested in two dimensions and for radial solutions by the result of [32] (see also [60]). …”

“…In this Section, we come back on the result of [31], [32], Theorem 2.5, that expresses the type of blow-up as a chemotactic collapse for radial solutions in two dimensions. We explain why, for radially symmetric solutions, the system can be simplified and we prove a blow-up result which is close to exhibit the shape of the chemotactic collapse solution.…”

“…The most famous being Patlak, Keller & Segel model ( [55,39]) which is formed of parabolic or elliptic equations coupled through a drift term. This model is very successful for describing the aggregation of the population in a single point (chemotactic collapse in the terminology of [32]). For this reason it has given rise to an important literature and we refer to the survey [35] for complements.…”

On a competitive system under chemotactic effects with non-local terms

“…One of the first mathematical studies for parabolicelliptic systems is the work by Jager and Luckhaus [16], where a sub-and super-solutions method is applied to obtain finite-time blow-up in a two-dimensional domain. After [16], many authors have studied the question of blow-up for parabolic-elliptic chemotaxis systems, see for instance Nagai [22], Herrero et al [11], Biler [3] and the references therein for more details.…”