Boundedness vs. blow-up in a chemotaxis system

By Dirk Horstmann, Michael Winkler. 2005

1,037 citations recorded by the source corpus.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Blow-up, Concentration Phenomenon and Global Existence for the Keller–Segel Model in High Dimension (2010, neutral): “Therefore, we are not able to obtain here a local existence result without any smallness conditions, nor to improve the known smallness condition for the global existence when > 0 (see [21, 40] for the global existence under smallness conditions and [36, 51] for the existence vs.”
On the global existence of solutions to an aggregation model (2008, methodology): “we immediately obtain L1 estimate of the gradient of c. Consider now the following sequence (in a similar way as was done in [ 28 ])”
Dissipative Gradient Nonlinearities Prevent δ$\delta$ ‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models (2024, neutral): “For (14) we refer to [21], whereas for (15) we can invoke [20, Lemma 4.1] and the embedding W 1 , nq ( n − q and there is y 1 > 0 with the property that whenever y > y 1 for some t ∈ (0 , T ) one has that ϕ ( t, y ) ≤ 0 , then y ≤ max { y 1 , y (0) } on (0 , T ) . and the lower bound of the…”
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system (2011, background): “This type of reasoning, well-established in scalar parabolic problems ([25]) and also applicable in larger classes of quasilinear Keller-Segel models ([18], [19], [35]), evidently cannot give any information on the actual blow-up time, and hence cannot rule out the possibility that Tmax(u0, v0) = ∞.”