1. Bedrossian, J., Kim, I.C.: Global existence and finite time blow-up for critical Patlak-Keller-Segel models
with inhomogeneous diffusion. SIAM J. Math. Anal. 45, 934–964 (2013)
2. Biler, P.: Existence and nonexistence of solutions for a model of gravitational interaction of particles, III.
Colloq. Math. 68, 229–239 (1995)
3. Biler, P.: Local and global solvability of some parabolic systems modeling chemotaxis. Adv. Math. Sci.
Appl. 8, 715–743 (1998)
4. Biler, P., Cie´slak, T., Karch, G., Zienkiewicz, J.: Local criteria for blowup in two-dimensional chemotaxis
models. Discrete Contin. Dyn. Syst. 37, 1841–1856 (2017)
5. Biler, P., Karch, G., Zienkiewicz, J.: Large global-in-time solutions to a nonlocal model of chemotaxis.
Adv. Math. 330, 834–875 (2018)
123
Finite time blow up and concentration ...
Page 33 of 34
47
6. Biler, P., Nadzieja, T.: Existence and nonexistence of solutions for a model of gravitational interactions
of particles I. Colloq. Math. 66, 319–334 (1994)
7. Biler, P., Nadzieja, T.: A nonlocal singular parabolic problem modeling gravitational interaction of particles. Adv. Diff. Equ. 3, 177–197 (1998)
8. Biler, P., Nadzieja, T., Stanczy, R.: Nonisothermal systems of self-attracting Fermi-Dirac particles. Banach
Center Pulb. 66, 61–78 (2004)
9. Bournaveas, N., Calvez, V.: The one-dimensional Keller-Segel model with fractional diffusion of cells.
Nonlinearity 23, 923–935 (2010)
10. Blanchet, A., Carrillo, J.A., Laurençot, P.: Critical mass for a Patlak-Keller-Segel model with degenerate
diffusion in higher dimensions. Calc. Var. P.D.E. 35, 133–168 (2009)
11. Calvez, V., Corrias, L., Ebde, M.: Blow-up, concentration phenomenon and global existence for the
Keller-Segel model in high dimension. Comm. Partial Differ. Equ. 37, 561–584 (2012)
12. Chen, L., Liu, J.G., Wang, J.: Multidimensional degenerate Keller-Segel system with critical diffusion
exponent 2n/(n + 2)∗ . SIAM J. Math. Anal. 44(2), 1077–1102 (2012)
13. Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis system in
hight space dimensions. Milan J. Math. 72, 1–28 (2004)
14. Feireisl, E., Laurençot, P.: Non-isothermal Smoluchowski-Poisson equations as a singular limit of the
Navier-Stokes-Fourier-Poisson system. J. Math. Pures Appl. 88, 325–349 (2007)
15. Gajewski, H., Gröger, K.: On the basic equations for carrier transport in semiconductors. J. Math. Anal.
Appl. 113, 12–35 (1986)
16. Gérard, P.: Description du défaut de compacité de l‘injection de Sobolev. ESAIM Control Optim. Calc.
Var. 3, 213–233 (1998)
17. Giga, Y.: Solutions for semilinear Parabolic equations in L p and regularity of weak solutions of the
Navier-Stokes system. J. Differ. Equ. 61, 186–212 (1986)
18. Herrero, M.A., Velázquez, J.J.L.: Singularity patterns in a chemotaxis model. Math. Ann. 306, 583–623
(1996)
19. Hmidi, T., Keraani, S.: Remarks on the blowup for the L 2 critical nonlinear Schrödinger equations. SIAM
J. Math. Anal. 38, 1035–1047 (2006)
20. Iwabuchi, T.: Global well-posedness for Keller-Segel system in Besov type spaces. J. Math. Anal. Appl.
379, 930–948 (2011)
21. Jüngel, A.: Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Model. Meth. Appl. Sci. 5, 497–518 (1995)
22. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26,
399–415 (1970)
23. Kimijima, A., Nakagawa, K., Ogawa, T.: Threshold of global behavior of solutions to a degenerate driftdiffusion system in between two critical exponents. Calc. Var. Partial Differ. Equ. 53, 441–472 (2015)
24. Kobayashi, T., Ogawa, T.: Fluid mechanical approximation to the degenerated drift-diffusion and chemotaxis equations in barotropic model. Indiana Univ. Math. J. 62(3), 1021–1054 (2013)
25. Kozono, H., Sugiyama, Y., Yahagi, Y.: Existence and uniqueness theorem on weak solutions to the
parabolic-elliptic Keller-Segel system. J. Differ. Equ. 253, 2295–2313 (2012)
26. Kurokiba, M., Nagai, T., Ogawa, T.: The uniform boundedness of the radial solution for drift-diffusion
system. Comm. Pure Appl. Anal. 5, 97–106 (2006)
27. Kurokiba, M., Ogawa, T.: Finite time blow-up of the solution for a nonlinear parabolic equation of
drift-diffusion type. Differ. Integral Equ. 16, 427–452 (2003)
28. Kurokiba, M., Ogawa, T.: L p well-posedness of the for the drift-diffusion system arising from the semiconductor device simulation. J. Math. Anal. Appl. 342, 1052–1067 (2008)
29. Kurokiba, M., Ogawa, T.: Two dimensional drift-diffusion system in a critical weighted space. Differ.
Integral Equ. 28(7–8), 753–776 (2015)
30. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact
case I. Ann. Inst. H. Poincaré Anal. Non Lin’eaire 1, 109–145 (1984)
31. Merle, F., Tsutsumi, Y.: L 2 concentration of blowup solutions for the nonlinear Schrödinger equation
with critical power nonlinearity. J. Differ. Equ. 84, 1035–1047 (1990)
32. Mock, M.S.: An initial value problem from semiconductor device theory, SIAM. J. Math. 5, 597–612
(1974)
33. Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxissystem. Adv. Math. Sci. Appl. 5,
581–601 (1995)
34. Nagai, T.: Blowup of non-radial solutions to parabolic-elliptic systems modeling chemotaxis in twodimensional domains. J. Inequal. Appl. 6, 37–55 (2001)
35. Nagai, T., Ogawa, T.: Brezis-Merle inequalities and application to the global existence of the Keller-Segel
equations. Comm. Contemporary Math. 13(5), 795–812 (2011)
123
47
Page 34 of 34
T. Ogawa et al.
36. Nagai, T., Ogawa, T.: Global existence of solutions to a parabolic-elliptic system of drift-diffusion type
in R2 , Funkcial. Ekvac. 59, No. 2 (2016),
37. Nagai, T., Senba, T., Suzuki, T.: Chemotactic collapse in a parabolic system of mathematical biology.
Hiroshima J. Math. 30, 463–497 (2000)
38. Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger-Moser inequality to a parabolic system
of chemotaxis. Funkcial. Ekvac. 40(3), 411–433 (1997)
39. Nawa, H.: Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation
with critical power. Comm. Pure Appl. Math. 52, 193–270 (1999)
40. Ogawa, T.: The degenerate drift-diffusion system with the Sobolev critical exponent. Disc. Contin. Dyn.
Syst. Ser S 4, 875–886 (2011)
41. Ogawa, T., Wakui, H.: Non-uniform bound and finite time blow up for solutions to a drift-diffusion
equation in higher dimensions. Anal. Appl. 14, 145–183 (2016)
42. Ogawa, T., Wakui, H.: Finite time blow up and non-uniform bound for solutions to a degenerate driftdiffusion equation with the mass critical exponent under non-weight condition. Manuscr. Math. 159,
475–509 (2019)
43. Senba, T.: Blowup behavior of radial solutions to Jäger-Luckhaus system in high dimensional domain.
Funkcilaj Ekvac. 48, 247–271 (2005)
44. Senba, T.: Blowup in infinite time of radial solutions to parabolic-elliptic system in high-dimensional
Euclidean spaces. Nonlinear Anal. 70, 2549–2562 (2009)
45. Sugiyama, Y.: Application of the best constant of the Sobolev inequality to degenerate Keller-Segel
models. Adv. Differ. Equ. 12, 121–144 (2007)
46. Suzuki, T., Takahashi, R.: Degenerate parabolic equations with critical exponent derived from the kinetic
theory II, Blowup threshold. Differ. Integral Equ 22, 1153–1172 (2009)
47. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)
48. Tsutsumi, Y.: Rate of L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with
critical power. Nonlinear Anal. 15, 719–724 (1990)
49. Wakui, H.: The rate of concentration for the radially symmetric solution to a degenerate drift-diffusion
equation with the mass critical exponent. Arch. Math. (Basel) 111, 535–548 (2018)
50. Weissler, F.: Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math.
38, 29–40 (1981)
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional affiliations.
123
...
論文の公開元へ
