메뉴 건너뛰기




Volumn 35, Issue 2, 2009, Pages 133-168

Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions

Author keywords

[No Author keywords available]

Indexed keywords


EID: 60449088258     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-008-0200-7     Document Type: Article
Times cited : (201)

References (41)
  • 1
    • 31744443874 scopus 로고    scopus 로고
    • Gradient flows in metric spaces and in the space of probability measures
    • Birkhäuser 2005
    • Ambrosio, L.A., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, Birkhäuser (2005)
    • Lectures in Mathematics
    • Ambrosio, L.A.1
  • 2
    • 0032368532 scopus 로고    scopus 로고
    • Long-wave instabilities and saturation in thin film equations
    • A.L. Bertozzi M.C. Pugh 1998 Long-wave instabilities and saturation in thin film equations Comm. Pure Appl. Math. 51 625 661
    • (1998) Comm. Pure Appl. Math. , vol.51 , pp. 625-661
    • Bertozzi, A.L.1    Pugh, M.C.2
  • 3
    • 0039623018 scopus 로고    scopus 로고
    • Finite-time blow-up of solutions of some long-wave unstable thin film equations
    • A.L. Bertozzi M.C. Pugh 2000 Finite-time blow-up of solutions of some long-wave unstable thin film equations Indiana Univ. Math. J. 49 1323 1366
    • (2000) Indiana Univ. Math. J. , vol.49 , pp. 1323-1366
    • Bertozzi, A.L.1    Pugh, M.C.2
  • 4
    • 33747147890 scopus 로고    scopus 로고
    • The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane
    • P. Biler G. Karch P. Laurençot T. Nadzieja 2006 The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane Math. Methods Appl. Sci. 29 1563 1583
    • (2006) Math. Methods Appl. Sci. , vol.29 , pp. 1563-1583
    • Biler, P.1    Karch, G.2    Laurençot, P.3    Nadzieja, T.4
  • 5
    • 55549135711 scopus 로고    scopus 로고
    • Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model
    • A. Blanchet V. Calvez J.A. Carrillo 2008 Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model SIAM J. Numer. Anal. 46 691 721
    • (2008) SIAM J. Numer. Anal. , vol.46 , pp. 691-721
    • Blanchet, A.1    Calvez, V.2    Carrillo, J.A.3
  • 6
    • 52349087894 scopus 로고    scopus 로고
    • Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model
    • A. Blanchet J.A. Carrillo N. Masmoudi 2008 Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model Comm. Pure Appl. Math. 61 1449 1481
    • (2008) Comm. Pure Appl. Math. , vol.61 , pp. 1449-1481
    • Blanchet, A.1    Carrillo, J.A.2    Masmoudi, N.3
  • 7
    • 33645680010 scopus 로고    scopus 로고
    • Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
    • electronic
    • Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 44 (2006), 32 pp (electronic)
    • (2006) Electron. J. Differ. Equ. , vol.44
    • Blanchet, A.1    Dolbeault, J.2    Perthame, B.3
  • 8
    • 33746377577 scopus 로고    scopus 로고
    • Volume effects in the Keller-Segel model: Energy estimates preventing blow-up
    • V. Calvez J.A. Carrillo 2006 Volume effects in the Keller-Segel model: energy estimates preventing blow-up J. Math. Pures Appl. 86 155 175
    • (2006) J. Math. Pures Appl. , vol.86 , pp. 155-175
    • Calvez, V.1    Carrillo, J.A.2
  • 10
    • 18044401055 scopus 로고    scopus 로고
    • Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities
    • J.A. Carrillo A. Jüngel P.A. Markowich G. Toscani A. Unterreiter 2001 Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities Monatsh. Math. 133 1 82
    • (2001) Monatsh. Math. , vol.133 , pp. 1-82
    • Carrillo, J.A.1    Jüngel, A.2    Markowich, P.A.3    Toscani, G.4    Unterreiter, A.5
  • 11
    • 1542399709 scopus 로고    scopus 로고
    • Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates
    • J.A. Carrillo R.J. McCann C. Villani 2003 Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates Rev. Mat. Iberoamericana 19 1 48
    • (2003) Rev. Mat. Iberoamericana , vol.19 , pp. 1-48
    • Carrillo, J.A.1    Villani, C.2
  • 12
    • 32544445429 scopus 로고    scopus 로고
    • Contractions in the 2-Wasserstein length space and thermalization of granular media
    • J.A. Carrillo R.J. McCann C. Villani 2006 Contractions in the 2-Wasserstein length space and thermalization of granular media Arch. Ration. Mech. Anal. 179 217 263
    • (2006) Arch. Ration. Mech. Anal. , vol.179 , pp. 217-263
    • Carrillo, J.A.1    Villani, C.2
  • 13
    • 0003277699 scopus 로고    scopus 로고
    • 1-decay of solutions of the porous medium equation to self-similarity
    • 1-decay of solutions of the porous medium equation to self-similarity Indiana Univ. Math. J. 49 113 142
    • (2000) Indiana Univ. Math. J. , vol.49 , pp. 113-142
    • Carrillo, J.A.1    Toscani, G.2
  • 14
    • 5444256656 scopus 로고    scopus 로고
    • Semilinear Schrödinger equations
    • New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence 2003
    • Cazenave, T.: Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (2003)
    • Courant Lecture Notes in Mathematics , vol.10
    • Cazenave, T.1
  • 15
    • 3342973415 scopus 로고    scopus 로고
    • Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions
    • P.-H. Chavanis C. Sire 2004 Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions Phys. Rev. E. 69 016116
    • (2004) Phys. Rev. E. , vol.69 , pp. 016116
    • Chavanis, P.-H.1    Sire, C.2
  • 16
    • 26844499863 scopus 로고    scopus 로고
    • Global solutions of some chemotaxis and angiogenesis systems in high space dimensions
    • L. Corrias B. Perthame H. Zaag 2004 Global solutions of some chemotaxis and angiogenesis systems in high space dimensions Milan J. Math. 72 1 28
    • (2004) Milan J. Math. , vol.72 , pp. 1-28
    • Corrias, L.1    Perthame, B.2    Zaag, H.3
  • 17
    • 18644382851 scopus 로고    scopus 로고
    • Optimal critical mass in the two-dimensional Keller-Segel model in R2
    • J. Dolbeault B. Perthame 2004 Optimal critical mass in the two-dimensional Keller-Segel model in R2 C. R. Math. Acad. Sci. Paris 339 611 616
    • (2004) C. R. Math. Acad. Sci. Paris , vol.339 , pp. 611-616
    • Dolbeault, J.1    Perthame, B.2
  • 18
    • 34250271532 scopus 로고
    • Symmetry and related properties via the maximum principle
    • B. Gidas W.-M. Ni L. Nirenberg 1979 Symmetry and related properties via the maximum principle Comm. Math. Phys. 68 209 243
    • (1979) Comm. Math. Phys. , vol.68 , pp. 209-243
    • Gidas, B.1    Ni, W.-M.2    Nirenberg, L.3
  • 20
    • 4744373150 scopus 로고    scopus 로고
    • From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. i
    • D. Horstmann 2003 From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I Jahresber. Deutsch. Math. Verein. 105 103 165
    • (2003) Jahresber. Deutsch. Math. Verein. , vol.105 , pp. 103-165
    • Horstmann, D.1
  • 21
    • 84966208077 scopus 로고
    • On explosions of solutions to a system of partial differential equations modelling chemotaxis
    • W. Jäger S. Luckhaus 1992 On explosions of solutions to a system of partial differential equations modelling chemotaxis Trans. Am. Math. Soc. 329 819 824
    • (1992) Trans. Am. Math. Soc. , vol.329 , pp. 819-824
    • Jäger, W.1    Luckhaus, S.2
  • 22
    • 0014748565 scopus 로고
    • Initiation of slide mold aggregation viewed as an instability
    • E.F. Keller L.A. Segel 1970 Initiation of slide mold aggregation viewed as an instability J. Theor. Biol. 26 399 415
    • (1970) J. Theor. Biol. , vol.26 , pp. 399-415
    • Keller, E.F.1    Segel, L.A.2
  • 23
    • 15844396394 scopus 로고    scopus 로고
    • Preventing blow-up in a chemotaxis model
    • R. Kowalczyk 2005 Preventing blow-up in a chemotaxis model J. Math. Anal. Appl. 305 566 588
    • (2005) J. Math. Anal. Appl. , vol.305 , pp. 566-588
    • Kowalczyk, R.1
  • 24
    • 0000298073 scopus 로고
    • Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
    • E.H. Lieb 1983 Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities Ann. Math. (2) 118 349 374
    • (1983) Ann. Math. (2) , vol.118 , pp. 349-374
    • Lieb, E.H.1
  • 25
    • 0242648945 scopus 로고    scopus 로고
    • Analysis
    • 2nd edn. American Mathematical Society, Providence
    • Lieb, E.H., Loss, M.: Analysis. In: Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)
    • (2001) Graduate Studies in Mathematics , vol.14
    • Lieb, E.H.1    Loss, M.2
  • 26
    • 0000539356 scopus 로고
    • The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics
    • E.H. Lieb H.-T. Yau 1987 The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics Comm. Math. Phys. 112 147 174
    • (1987) Comm. Math. Phys. , vol.112 , pp. 147-174
    • Lieb, E.H.1    Yau, H.-T.2
  • 27
    • 85030707196 scopus 로고
    • The concentration-compactness principle in the calculus of variations. the locally compact case. i
    • P.-L. Lions 1984 The concentration-compactness principle in the calculus of variations. The locally compact case. I Ann. Inst. H. Poincaré Anal. Non Linéaire 1 109 145
    • (1984) Ann. Inst. H. Poincaré Anal. Non Linéaire , vol.1 , pp. 109-145
    • Lions, P.-L.1
  • 28
    • 0031161793 scopus 로고    scopus 로고
    • A convexity principle for interacting gases
    • 1
    • R.J. McCann 1997 A convexity principle for interacting gases Adv. Math. 128 1 153 179
    • (1997) Adv. Math. , vol.128 , pp. 153-179
    • McCann, R.J.1
  • 29
    • 2942586742 scopus 로고    scopus 로고
    • 2 critical nonlinear Schrödinger equation
    • 2 critical nonlinear Schrödinger equation Invent. Math. 156 565 672
    • (2004) Invent. Math. , vol.156 , pp. 565-672
    • Merle, F.1    Raphaël, P.2
  • 30
    • 60449112899 scopus 로고    scopus 로고
    • Decay and asymptotic behavior of solutions of the Keller-Segel system of degenerate and nondegenerate type, Self-similar solutions of nonlinear PDE
    • Polish Acad. Sci., Warsaw
    • Ogawa, T.: Decay and asymptotic behavior of solutions of the Keller-Segel system of degenerate and nondegenerate type, Self-similar solutions of nonlinear PDE, pp. 161-184, Banach Center Publ., 74, Polish Acad. Sci., Warsaw (2006)
    • (2006) Banach Center Publ. , vol.74 , pp. 161-184
    • Ogawa, T.1
  • 31
    • 0001560970 scopus 로고    scopus 로고
    • The geometry of dissipative evolution equations: The porous medium equation
    • F. Otto 2001 The geometry of dissipative evolution equations: the porous medium equation Comm. Partial Differ. Equ. 26 101 174
    • (2001) Comm. Partial Differ. Equ. , vol.26 , pp. 101-174
    • Otto, F.1
  • 32
    • 51249193541 scopus 로고
    • Random walk with persistence and external bias
    • C.S. Patlak 1953 Random walk with persistence and external bias Bull. Math. Biophys. 15 311 338
    • (1953) Bull. Math. Biophys. , vol.15 , pp. 311-338
    • Patlak, C.S.1
  • 33
    • 31644436781 scopus 로고    scopus 로고
    • Selfsimilar blowup of unstable thin-film equations
    • D. Slepčev M.C. Pugh 2005 Selfsimilar blowup of unstable thin-film equations Indiana Univ. Math. J. 54 1697 1738
    • (2005) Indiana Univ. Math. J. , vol.54 , pp. 1697-1738
    • Slepčev, D.1    Pugh, M.C.2
  • 34
    • 34547418914 scopus 로고    scopus 로고
    • Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems
    • Y. Sugiyama 2006 Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems Differ. Integral Equ. 19 841 876
    • (2006) Differ. Integral Equ. , vol.19 , pp. 841-876
    • Sugiyama, Y.1
  • 35
    • 43049127610 scopus 로고    scopus 로고
    • Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models
    • Y. Sugiyama 2007 Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models Adv. Differ. Equ. 12 121 144
    • (2007) Adv. Differ. Equ. , vol.12 , pp. 121-144
    • Sugiyama, Y.1
  • 37
    • 16244371027 scopus 로고    scopus 로고
    • Swarming patterns in a two-dimensional kinematic model for biological groups
    • C.M. Topaz A.L. Bertozzi 2004 Swarming patterns in a two-dimensional kinematic model for biological groups SIAM J. Appl. Math. 65 152 174
    • (2004) SIAM J. Appl. Math. , vol.65 , pp. 152-174
    • Topaz, C.M.1    Bertozzi, A.L.2
  • 38
    • 33748692474 scopus 로고    scopus 로고
    • A nonlocal continuum model for biological aggregation
    • C.M. Topaz A.L. Bertozzi M.A. Lewis 2006 A nonlocal continuum model for biological aggregation Bull. Math. Biol. 68 1601 1623
    • (2006) Bull. Math. Biol. , vol.68 , pp. 1601-1623
    • Topaz, C.M.1    Bertozzi, A.L.2    Lewis, M.A.3
  • 39
    • 0041473959 scopus 로고
    • Nonlinear Schrödinger equations and sharp interpolation estimates
    • M.I. Weinstein 1983 Nonlinear Schrödinger equations and sharp interpolation estimates Comm. Math. Phys. 87 567 576
    • (1983) Comm. Math. Phys. , vol.87 , pp. 567-576
    • Weinstein, M.I.1
  • 40
    • 0001295566 scopus 로고
    • On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations
    • M.I. Weinstein 1986 On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations Comm. Partial Differ. Equ. 11 545 565
    • (1986) Comm. Partial Differ. Equ. , vol.11 , pp. 545-565
    • Weinstein, M.I.1
  • 41
    • 0000440099 scopus 로고
    • Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Ž. Èksper
    • V.E. Zakharov A.B. Shabat 1971 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Ž. Èksper Teoret. Fiz. 61 118 134
    • (1971) Teoret. Fiz. , vol.61 , pp. 118-134
    • Zakharov, V.E.1    Shabat, A.B.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.