Partial differential equations
Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source
Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 913-917.

We consider the quasilinear parabolic–elliptic chemotaxis system

{ut=(D(u)uχuv)+g(u),xΩ,t>0,0=Δvv+u,xΩ,t>0,
under homogeneous Neumann boundary conditions in a smooth bounded domain ΩRn,n1. We assume that the functions D and g are smooth and satisfy
D(s)>0fors0,D(s)CDsm1fors>0,g(0)0,g(s)absγ,s>0
with some constants CD>0,m1,a0,b>0 and γ>2.

We prove that the classical solutions to the above system are uniformly in-time-bounded without any restrictions on m and b. This result extends one of the recent results by Wang et al. (2014) [16], which assert the boundedness of solutions for γ>2 under the condition b>b with b=0 for m22n and b=(2m)n2(2m)nχ for m<22n.

Nous considérons le système quasi linéaire, parabolique–elliptique, de chimiotaxie

{ut=(D(u)uχuv)+g(u),xΩ,t>0,0=Δvv+u,xΩ,t>0,
avec des conditions au bord homogènes de Neumann, dans un domaine lisse, borné ΩRn, n1. Nous supposons que les fonctions D et
D(s)>0pours0,D(s)CDsm1pours>0,
g(0)0,g(s)absγ,s>0
pour certaines constantes CD>0, m1, a0, b>0 et γ>2.

Nous démontrons que les solutions classiques du système ci-dessus sont uniformément bornées en temps, sans restriction sur m et b. Ceci étend un résultat récent de Wang et al. (2014) [16], qui borne les solutions pour γ>2 sous la condition b>b, où b=0 si m22n et b=(2m)n2(2m)nχ si m<22n.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.08.006
Khelghati, Ali 1; Baghaei, Khadijeh 1

1 Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697, Tehran, Iran
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Khelghati, Ali; Baghaei, Khadijeh. Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source. Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 913-917. doi : 10.1016/j.crma.2015.08.006. http://www.numdam.org/articles/10.1016/j.crma.2015.08.006/

[1] Alikakos, N.D. Lp bounds of solutions of reaction–diffusion equations, Commun. Partial Differ. Equ., Volume 4 (1979), pp. 827-868

[2] Cao, X. Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., Volume 412 (2014), pp. 181-188

[3] Cao, X.; Zheng, S. Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, Math. Methods Appl. Sci., Volume 37 (2014), pp. 2326-2330

[4] Hillen, T.; Painter, K.J. A user's guide to PDE models for chemotaxis, J. Math. Biol., Volume 58 (2009), pp. 183-217

[5] Horstmann, D.; Wang, G. Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., Volume 12 (2001), pp. 159-177

[6] Horstmann, D.; Winkler, M. Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., Volume 215 (2005), pp. 52-107

[7] Keller, E.F.; Segel, L.A. Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., Volume 26 (1970), pp. 399-415

[8] Lankeit, J. Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ., Volume 258 (2015), pp. 1158-1191

[9] Lankeit, J. Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst., Ser. B, Volume 20 (2015), pp. 1499-1527

[10] Nagai, T.; Senba, T.; Yoshida, K. Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Volume 40 (1997), pp. 411-433

[11] Osaki, K.; Yagi, A. Finite dimensional attractors for one-dimensional Keller–Segel equations, Funkc. Ekvacioj, Volume 44 (2001), pp. 441-469

[12] Osaki, K.; Yagi, A. Global existence of a chemotaxis-growth system in R2, Adv. Math. Sci. Appl., Volume 12 (2002), pp. 587-606

[13] Tao, Y.; Winkler, M. Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Differ. Equ., Volume 252 (2012), pp. 692-715

[14] Tello, J.I.; Winkler, M. A chemotaxis system with logistic source, Commun. Partial Differ. Equ., Volume 32 (2007), pp. 849-877

[15] Temam, R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997

[16] Wang, L.; Mu, C.; Zheng, P. On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Differ. Equ., Volume 256 (2014), pp. 1847-1872

[17] Winkler, M. Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., Volume 348 (2008), pp. 708-729

[18] Winkler, M. Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differ. Equ., Volume 248 (2010), pp. 2889-2905

[19] Winkler, M. Does ‘volume-filling effect’ always prevent chemotactic collapse?, Math. Methods Appl. Sci., Volume 33 (2010), pp. 12-24

[20] Winkler, M. Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 1516-1537

[21] Winkler, M. Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., Volume 384 (2011), pp. 261-272

[22] Winkler, M. Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., Volume 100 (2013), pp. 748-767

[23] Winkler, M. Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differ. Equ., Volume 257 (2014), pp. 1056-1077

[24] Winkler, M. How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., Volume 24 (2014), pp. 809-855

[25] Winkler, M.; Djie, K.C. Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA, Volume 72 (2010), pp. 1044-1064

[26] Zheng, P.; Mu, C.; Hu, X. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., Volume 35 (2015), pp. 2299-2323

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