2008
DOI: 10.1016/j.jmaa.2008.01.005
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On the global existence of solutions to an aggregation model

Abstract: In this paper we consider a reaction-diffusion-chemotaxis aggregation model of Keller-Segel type with a nonlinear, degenerate diffusion. Assuming that the diffusion function f (n) takes values sufficiently large, i.e. takes values greater than the values of a power function with sufficiently high power (f (n) δn p for all n > 0, where δ > 0 is a constant), we prove global-in-time existence of weak solutions. Since one of the main features of Keller-Segel type models is the possibility of blow-up of solutions i… Show more

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Cited by 174 publications
(68 citation statements)
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“…With the help of (1.2) and the boundedness of u i (i = 1, 2) on Ω × (0, T ), we can find positive constants c 15 , c 16 , c 17 , c 18 , c 19 , c 20 and…”
Section: Preliminariesmentioning
confidence: 99%
See 1 more Smart Citation
“…With the help of (1.2) and the boundedness of u i (i = 1, 2) on Ω × (0, T ), we can find positive constants c 15 , c 16 , c 17 , c 18 , c 19 , c 20 and…”
Section: Preliminariesmentioning
confidence: 99%
“…In fact, the two behaviors (boundedness and blow-up) of solutions strongly depend on the space dimension and the total mass of cells (see solutions strongly depend on the space dimension [2,[7][8][9][10]13,14,16,24,27,29,41]). However, it is shown by some recent studies that the volume-filling or prevention of overcrowding (see [3,11,12,17,45]), the nonlinear diffusion (see [8,18,19,30]) and the logistic damping (see [25,32,38,39,42,43]) may prevent the blow-up of solutions.…”
Section: Introductionmentioning
confidence: 99%
“…When the diffusion function D(U) takes values sufficiently large, i.e. takes values greater than the values of a power function with sufficiently high power, in [5], Kowalczyk and Szymańska proved that global-in-time existence of weak solutions. In addition, the uniqueness of solutions was given provided that some higher regularity condition on solutions is known a priori.…”
Section: Introductionmentioning
confidence: 99%
“…The interesting feature of KS type models is the possibility of blow-up of solutions in finite time, which strongly depends on the space dimension and the initial mass (see [4,[8][9][10]15,[18][19][20]29,30,32], for instance). However, some recent studies show that the volume-filling or prevention of overcrowding (see [5,16,17,45]), the nonlinear diffusion (see [9,22,33]), and the logistic damping (see [36,[38][39][40][41]44]) may prevent the blow-up of solutions. We should also point out that the mathematical analysis of the 3 × 3 chemotaxis-haptotaxis model differs from that of the 2 × 2 chemotaxis model due to some technique reasons (see [39,Section 1]).…”
Section: Introductionmentioning
confidence: 99%