Grow-up rate and refined asymptotics for a two-dimensional patlak-keller-segel model in a disk

SIAM Journal on Mathematical Analysis

Volumn 40, Issue 5, 2009, Pages 1852-1881

  Kavallaris, Nikos I ^a   Souplet, Philippe ^b  

^a UNIVERSITY OF THE AEGEAN   (Greece)

^b UNIVERSITÉ PARIS 13   (France)

Author keywords

Chemotaxis system; Critical mass; Grow up; Sub supersolutions

Indexed keywords

ASYMPTOTIC ESTIMATES; ASYMPTOTICS; CHEMOTAXIS SYSTEM; CRITICAL MASS; CRITICAL VALUE; GROW-UP; INFINITE TIME; PATLAK-KELLER-SEGEL MODEL; SUB-/SUPERSOLUTIONS; TOTAL MASS;

ASYMPTOTIC ANALYSIS; REFINING;

BIOCHEMISTRY;

EID: 70350090848     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/080722229     Document Type: Article

References (51)

1. F. BAVAUD, Equilibrium properties of the Vlasov functional: The generalized Poisson-Boltzmann-Emden equation, Rev. Modern Phys., 63 (1991), pp. 129-149.
2. P. BILER, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Colloq. Math., 68 (1995), pp. 229-239.
3. P. BILER, Local and global solvability of some parabolic systems modeling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), pp. 715-743.
4. P. BILER, D. HILHORST, AND T. NADZIEJA, Existence and nonexistence of solutions for a model of gravitational interaction of particles II, Colloq. Math., 67 (1994), pp. 297-308.
5. P. BILER, G. KARCH, P. LAURENÇOT, AND T. NADZIEJA, The 8-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol. Methods Nonlinear Anal., 27 (2006), pp. 133-144.
6. P. BILER, G. KARCH, P. LAURENÇOT, AND T. NADZIEJA, The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci., 29 (2006), pp. 1563-1583.
7. P. BILER AND T. NADZIEJA, Existence and nonexistence of solutions for a model of gravitational interaction of particles II, Colloq. Math., 66 (1994), pp. 319-334.
8. P. BILER AND T. NADZIEJA, Growth and accretion of mass in an astrophysical model II, Appl. Math., 23 (1995), pp. 351-361.
9. P. BILER AND T. NADZIEJA, A nonlocal singular parabolic problem modelling gravitational interaction of particles, Adv. Differential Equations, 3 (1998), pp. 177-197.
10. A. BLANCHET, J.A. CARRILLO, AND N. MASMOUDI, Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model, Comm. Pure Appl. Math., 61 (2008), pp. 1449-1481.
11. A. BLANCHET, J. DOLBEAULT, AND B. PERTHAME, Two dimensional Keller-Segel model: Optimal critical mass and qualitative properties of solutions, Electron. J. Differential Equations, 44 (2006), pp. 1-32.
12. M.P. BRENNER, P. CONSTANTIN, L.P. KADANOFF, A. SCHENKEL, AND S.C. VENKATARAMANI, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), pp. 1071-1098.
13. S. CHANDRASEKHAR, An Introduction to the Study of Stellar Structure, Dover, New York, 1967.
14. S. CHILDRESS AND J.K. PERCUS, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), pp. 217-237.
15. M. CHIPOT and F.B WELSSLER, Some blowup results for a nonliner parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), pp. 886-907.
16. M. DOI AND S.F. EDWARDS, The Theory of Polymer Dynamics, Clarendon Press, Oxford, UK, 1986.
17. 2, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 611-616.
18. J.W. DOLD, V.A. GALAKTIONOV, A.A. LACEY, AND J.L. VÁZQUEZ, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4), 26 (1998), pp. 663-687.
19. E. FEIREISL, H. PETZELTOVA, AND P. LAURENÇOT, On convergence to equilibria for the Keller- Segel chemotaxis model, J. Differential Equations, 236 (2007), pp. 551-569.
20. M. FILA, M. WINKLER, AND E. YANAGIDA, Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Differential Equations, 205 (2004), pp. 365-389. (Pubitemid 39150811)
21. M. FILA, J.R. KING, M. WINKLER, AND E. YANAGIDA, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Differential Equations, 228 (2006), pp. 339-356.
22. V.A. GALAKTIONOV AND J.R. KING, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), pp. 199-233.
23. I.A. GUERRA AND M.A. PELETIER, Self-similar blow-up for a diffusion-attraction problem, Nonlinearity, 17 (2004), pp. 2137-2162.
24. M.A. HERRERO, The mathematics of chemotaxis, in Handbook of Differential Equations: Evolutionary Equations, Vol.3, Elsevier, New York, 2007, pp. 137-193.
25. M.A. HERRERO AND J.J.L. VELÁZQUEZ, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), pp. 583-623.
26. M.A. HERRERO AND J.J.L. VELÁZQUEZ, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), pp. 177-194.
27. M.A. HERRERO AND J.J.L. VELÁZQUEZ J A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4), 24 (1997), pp. 633-683.
28. D. HORSTMANN, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 105 (2003), pp. 103-165.
29. D. HORSTMANN, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 106 (2004), pp. 51-69.
30. W. JÄGER AND S. LUCKHAUS, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc, 329 (1992), pp. 819-824.
31. N.I. KAVALLARIS AND T. SUZUKI, On the finite-time blow-up of a non-local parabolic equation describing chemotaxis, Differential Integral Equations, 20 (2007), pp. 293-308.
32. E.F. KELLER AND L.A. SEGEL, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), pp. 399-415.
33. A.A. LACEY AND D. TZANETIS, Global existence and convergence to a singular steady state for a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), pp. 289-305.
34. N. MIZOGUCHI, Growup of solutions for a semilinear heat equation with supercritical nonlinearity, J. Differential Equations, 227 (2006), pp. 652-669. (Pubitemid 44234400)
35. T. NAGAI, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), pp. 1-21.
36. T. NAGAI, Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), pp. 37-55.
37. T. NAGAI, T. SENBA, AND K. YOSHIDA, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), pp. 411-433.
38. V. NANJUNDIAH, Chemotaxis, signal relaying, and aggregation morphology, J. Theor. Biol., 42 (1973), pp. 63-105.
39. K. OHTSUKA, T. SENBA, AND T. SUZUKI, Blowup in infinite time in the simplified system of chemotaxis, Adv. Math. Sci. Appl., 17 (2007), pp. 445-472.
40. C.S. PATLAK, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), pp. 311-338.
41. B. PERTHAME, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), pp. 539-564.
42. P. POLAČIK AND E. YANAGIDA, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., 327 (2003), pp. 745-771.
43. P. QUITTNER AND PH. SOUPLET, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007.
44. T. SENBA AND T. SUZUKI, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), pp. 21-50.
45. T. SENBA, Blowup in infinite time of radial solutions for a parabolic-elliptic system in high dimensional Euclidean spaces, Nonlinear Anal., to appear.
46. C. SIRE AND P.-H. CHAVANIS, Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions, Phys. Rev. E (3), 66 (2002), 046133.
47. Ph. SOUPLET AND J.L. VÁZQUEZ, Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem, Discrete Contin. Dyn. Syst., 14 (2006), pp. 221-234.
48. T. SUZUKI, Free Energy and Self-Interacting Particles, Birkhäuser Boston, Boston, 2005.
49. G. WOLANSKY, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Ration. Mech. Anal., 119 (1992), pp. 355-391.
50. G. WOLANSKY, On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity, J. Anal. Math., 59 (1992), 251-272.
51. G. WOLANSKY, A critical parabolic estimate and application to nonlocal equations arising in chemotaxis, Appl. Anal., 66 (1997), 291-321.