zbMATH — the first resource for mathematics

Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions. (English) Zbl 1172.35035
The paper concerns qualitative properties of non-negative solutions for the Patlak-Keller-Segel system of dimension \(d \geq 3\) with homogeneous nonlinear diffusion. The diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. It is exhibited that the qualitative behavior of solutions is decided by the initial mass of the system, i.e. there is a sharp critical mass \(M_c\) such that free energy solutions exist globally for \(M \in (0, M_c]\) while there are finite time blowing-up solutions otherwise.
The main results of the work are as follows. If \(0\leq M\leq M_c\) then solutions exist globally in time, and there exists a radially symmetric compactly supported self-similar solution, although the authors could not show that it attracts all global solutions. If \(M=M_c\) then solutions exist globally in time, and there are infinitely many compactly supported stationary solutions. If \(M>M_c\) then it is proved that there exist solutions, corresponding to initial data with negative free energy, blowing up in finite time. However, the possibility that solutions with positive free energy may be global in time cannot be excluded.

35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Ambrosio, L.A., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, Birkhäuser (2005) · Zbl 1090.35002
[2] Bertozzi A.L., Pugh M.C.: Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51, 625–661 (1998) · Zbl 0916.35008 · doi:10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9
[3] Bertozzi A.L., Pugh M.C.: Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J. 49, 1323–1366 (2000) · Zbl 0978.35007 · doi:10.1512/iumj.2000.49.1887
[4] Biler, P., Karch, G., Laurençot, P., Nadzieja, T.: The 8\(\pi\)-problem for radially symmetric solutions of a chemotaxis model in the plane. Math. Methods Appl. Sci. 29, 1563–1583 (2006) · Zbl 1105.35131 · doi:10.1002/mma.743
[5] Blanchet, A., Calvez, V., Carrillo, J.A.: Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model. SIAM J. Numer. Anal. 46, 691–721 (2008) · Zbl 1205.65332 · doi:10.1137/070683337
[6] Blanchet A., Carrillo J.A., Masmoudi N.: Infinite time aggregation for the critical two-dimensional Patlak–Keller–Segel model. Comm. Pure Appl. Math. 61, 1449–1481 (2008) · Zbl 1155.35100 · doi:10.1002/cpa.20225
[7] 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) · Zbl 1112.35023
[8] Calvez V., Carrillo J.A.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86, 155–175 (2006) · Zbl 1116.35057 · doi:10.1016/j.matpur.2006.04.002
[9] Carlen E., Loss M.: Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on S n . Geom. Funct. Anal. 2, 90–104 (1992) · Zbl 0754.47041 · doi:10.1007/BF01895706
[10] Carrillo J.A., Jüngel A., Markowich P.A., Toscani G., Unterreiter A.: Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133, 1–82 (2001) · Zbl 0984.35027 · doi:10.1007/s006050170032
[11] Carrillo J.A., McCann R.J., Villani C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19, 1–48 (2003) · Zbl 1073.35127
[12] Carrillo J.A., McCann R.J., Villani C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179, 217–263 (2006) · Zbl 1082.76105 · doi:10.1007/s00205-005-0386-1
[13] Carrillo J.A., Toscani G.: Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113–142 (2000) · Zbl 0963.35098 · doi:10.1512/iumj.2000.49.1756
[14] 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) · Zbl 1055.35003
[15] Chavanis P.-H., Sire C.: Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions. Phys. Rev. E. 69, 016116 (2004) · doi:10.1103/PhysRevE.69.016116
[16] Corrias L., Perthame B., Zaag H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004) · Zbl 1115.35136 · doi:10.1007/s00032-003-0026-x
[17] Dolbeault, J., Perthame, B.: Optimal critical mass in the two-dimensional Keller–Segel model in \({\mathbb {R}^2}\) C. R. Math. Acad. Sci. Paris 339, 611–616 (2004) · Zbl 1056.35076
[18] Gidas B., Ni W.-M., Nirenberg L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125
[19] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 224, 2nd edn. Springer, Berlin (1983) · Zbl 0562.35001
[20] Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math. Verein. 105, 103–165 (2003) · Zbl 1071.35001
[21] Jäger,W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992) · Zbl 0746.35002 · doi:10.2307/2153966
[22] Keller E.F., Segel L.A.: Initiation of slide mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970) · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[23] Kowalczyk R.: Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305, 566–588 (2005) · Zbl 1065.35063 · doi:10.1016/j.jmaa.2004.12.009
[24] Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (2) 118, 349–374 (1983) · Zbl 0527.42011 · doi:10.2307/2007032
[25] Lieb, E.H., Loss, M.: Analysis. In: Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001) · Zbl 0966.26002
[26] Lieb E.H., Yau H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys. 112, 147–174 (1987) · Zbl 0641.35065 · doi:10.1007/BF01217684
[27] Lions P.-L.: 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)
[28] McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997) · Zbl 0901.49012 · doi:10.1006/aima.1997.1634
[29] Merle F., Raphaël P.: On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565–672 (2004) · Zbl 1067.35110 · doi:10.1007/s00222-003-0346-z
[30] 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) · Zbl 1133.35020
[31] Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26, 101–174 (2001) · Zbl 0984.35089 · doi:10.1081/PDE-100002243
[32] Patlak C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953) · Zbl 1296.82044 · doi:10.1007/BF02476407
[33] Slepčev D., Pugh M.C.: Selfsimilar blowup of unstable thin-film equations. Indiana Univ. Math. J. 54, 1697–1738 (2005) · Zbl 1091.35071 · doi:10.1512/iumj.2005.54.2569
[34] Sugiyama, Y.: 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) · Zbl 1212.35240
[35] Sugiyama, Y.: Application of the best constant of the Sobolev inequality to degenerate Keller–Segel models. Adv. Differ. Equ. 12, 121–144 (2007) · Zbl 1171.35068
[36] Sulem, C., Sulem, P.L.: The nonlinear Schrödinger equation. Applied Mathematical Sciences, vol. 139. Springer, New York (1999) · Zbl 0928.35157
[37] Topaz C.M., Bertozzi A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65, 152–174 (2004) · Zbl 1071.92048 · doi:10.1137/S0036139903437424
[38] Topaz C.M., Bertozzi A.L., Lewis M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68, 1601–1623 (2006) · Zbl 1334.92468 · doi:10.1007/s11538-006-9088-6
[39] Weinstein M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87, 567–576 (1983) · Zbl 0527.35023 · doi:10.1007/BF01208265
[40] Weinstein, M.I.: On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Comm. Partial Differ. Equ. 11, 545–565 (1986) · Zbl 0596.35022 · doi:10.1080/03605308608820435
[41] Zakharov V.E., Shabat A.B.: 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)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.