×

Ultra-contractivity for Keller-Segel model with diffusion exponent \(m>1-2/d\). (English) Zbl 1295.35271

Summary: This paper establishes the hyper-contractivity in \(L^\infty(\mathbb R^d)\) (it’s known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the diffusion exponent \(m>1-2/d\). The results show that for the supercritical and critical case \(1-2/d<m\leq 2-2/d\), if \(||U_0||_{d(2-m)/2}<C_{d,m}\) where \(C_{d,m}\) is a universal constant, then for any \(t>0\), \(||u(\cdot,t)||_{L^\infty(\mathbb R^d)}\) is bounded and decays as \(t\) goes to infinity. For the subcritical case \(m>2-2/d\), the solution \(u(\cdot,t)\in L^\infty(\mathbb R^d)\) with any initial data \(U_0\in L_+^1(\mathbb R^d)\) for any positive time.

MSC:

35K65 Degenerate parabolic equations
35B45 A priori estimates in context of PDEs
35J20 Variational methods for second-order elliptic equations
92C17 Cell movement (chemotaxis, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. D. Alikakos, \(L^p\) bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4, 827 (1979) · Zbl 0421.35009 · doi:10.1080/03605307908820113
[2] J. Bedrossian, Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models,, Comm. Math. Sci., 9, 1143 (2011) · Zbl 1282.35053 · doi:10.4310/CMS.2011.v9.n4.a11
[3] S. Bian, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent \(m > 0\),, Comm Math Phy., 323, 1017 (2013) · Zbl 1292.35061 · doi:10.1007/s00220-013-1777-z
[4] A. Blanchet, Infinite time aggregation for the critical Patlak-Keller-Segel model in \(\mathbbR^2\),, Comm. Pure Appl. Math., 61, 1449 (2008) · Zbl 1155.35100 · doi:10.1002/cpa.20225
[5] A. Blanchet, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Eletron. J. Differ. Equ. (2006) · Zbl 1112.35023
[6] M. P. Brenner, Diffusion, attraction and collapse,, Nonlinearity, 12, 1071 (1999) · Zbl 0942.35018 · doi:10.1088/0951-7715/12/4/320
[7] V. Calvez, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension,, Comm. Part. Diff. Eq., 37, 561 (2012) · Zbl 1255.35054 · doi:10.1080/03605302.2012.655824
[8] E. A. Carlen, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows,, Proc. Nat. Acad. USA, 107, 19696 (2010) · Zbl 1256.42028 · doi:10.1073/pnas.1008323107
[9] L. Corrias, Critical space for the parabolic-parabolic Keller-Segel model in \(\mathbbR^d\),, C. R. Acad. Sc. Paris, 342, 745 (2006) · Zbl 1097.35066 · doi:10.1016/j.crma.2006.03.008
[10] M. Del Pino, Nonlinear diffusions, hypercontractivity and the optimal \(L^p\)-Euclidean logarithmic Sobolev inequality,, J. Math. Anal. Appl., 293, 375 (2004) · Zbl 1058.35124 · doi:10.1016/j.jmaa.2003.10.009
[11] M. Herrero, Finite-time aggregation into a single point in a reaction-diffusion system,, Nonlinearity, 10, 1739 (1997) · Zbl 0909.35071 · doi:10.1088/0951-7715/10/6/016
[12] M. Herrero, Self-similar blow-up for a reaction-diffusion system,, J. Comp. Appl. Math., 97, 99 (1998) · Zbl 0934.35066 · doi:10.1016/S0377-0427(98)00104-6
[13] E. F. Keller, Initiation of slime mold aggregation viewed as an instability,, J. theor. Biol., 26, 399 (1970) · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[14] I. Kim, The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle,, SIAM J. Math. Anal., 44, 568 (2012) · Zbl 1261.35080 · doi:10.1137/110823584
[15] E. H. Lieb, <em>Analysis</em>,, Graduate Studies in Mathematics. V. 14 (2001) · doi:10.1080/13683500108667891
[16] B. Perthame, <em>Transport Equations in Biology</em>,, Birkhaeuser Verlag (2007) · Zbl 1185.92006
[17] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate keller-segel systems,, Diff. Int. Eqns., 19, 841 (2006) · Zbl 1212.35240
[18] Y. Sugiyama, Global existence and decay properties for a degenerate keller-segel model with a power factor in drift term,, J. Diff. Eqns., 227, 333 (2006) · Zbl 1102.35046 · doi:10.1016/j.jde.2006.03.003
[19] J. L. Vázquez, <em>Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type</em>,, Oxford Lecture Ser. Math. Appl. (2006) · Zbl 1113.35004
[20] J. L. Vázquez, <em>The Porous Medium Equation: Mathematical Theory</em>,, Oxford Mathematical Monographs. The Clarendon Press (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.