Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation

Mathematical Models and Methods in Applied Sciences

Volumn 19, Issue 6, 2009, Pages 939-967

  Ogawa, Takayoshi ^a   Yamamoto, Masakazu ^a  

^a TOHOKU UNIVERSITY   (Japan)

Author keywords

Asymptotic profiles.; Decay of solution; Drift diffusion system; Fractional order derivatives; Keller Segel equations; Large data global solutions

Indexed keywords

ASYMPTOTIC PROFILES.; DECAY OF SOLUTION; DRIFT-DIFFUSION SYSTEM; FRACTIONAL ORDER DERIVATIVES; KELLER-SEGEL EQUATIONS; LARGE DATA GLOBAL SOLUTIONS;

ASYMPTOTIC ANALYSIS; DIFFUSION; PARTIAL DIFFERENTIAL EQUATIONS;

MATHEMATICAL MODELS;

EID: 68249118543     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021820250900367X     Document Type: Article

References (41)

1. P. Biler and J. Dolbeault, Long time behavior of solutions to Nernst - Planck and Debye - Hückel drift-diffusion systems, Ann. Henri Poincaré 1 (2000) 461 - 472.
2. P. Biler, G. Karch and W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré, Analyse Non Liné aire 18 (2001) 613 - 637.
3. P. Biler and T. Nadzieja, A nonlocal singular parabolic problem modeling gravitational interaction of particles, Adv. Diff. Eqns. 3 (1998) 177 - 197.
4. P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math. 59 (1998) 845 - 869.
5. P. Biler and W. A. Woyczyński, General nonlocal diffusion-convection mean field models: Nonexistence of global solutions, Physica A 379 (2007) 523 - 533.
6. P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Meth. Appl. Sci. (2008), DOI: 10.1002/mma.1036.
7. A. Carpio, Large-time behavior in incompressible Navier - Stokes equation, SIAM J. Math. Anal. 27 (1996) 449 - 475.
8. T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations (Oxford Science Publications, 1998).
9. A. Córdoba and D. Córdoba, A pointwise estimate for fractionary derivatives with applications to partial differential equations, Proc. Natl. Acad. Sci. USA 100 (2003) 15316 - 15317 (electronic).
10. A. Cordoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys. 249 (2004) 511 - 528.
11. n, J. Funct. Anal. 100 (1991) 119 - 161.
12. C. Escudero, The fractional Keller - Segel model, Nonlinearity 19 (2006) 2909 - 2918.
13. W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Diff. Eqns. 123 (1995) 523 - 566.
14. Y. Fujigaki and T. Miyakawa, Asymptotic profiles of nonstationary incompressible Navier - Stokes flows in the whole space, SIAM J. Math. Anal. 33 (2001) 523 - 544.
15. M.-H. Giga and Y. Giga, Nonlinear Partial Differential Equations (in Japanese) (Kyoritsu Publ., 1999).
16. N. Hayashi, E. I. Kaikina and P. I. Naumkin, Asymptotic for fractional nonlinear heat equations, J. London Math. Soc. 72(2) (2005) 663 - 688.
17. W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc. 329 (1992) 819 - 824.
18. A. Jiingel, Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Mod. Meth. Appl. Sci. 5 (1995) 497 - 518.
19. M. Kato, Sharp asymptotics for a parabolic system of chemotaxis in one space dimension, preprint, Osaka University.
20. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier - Stokes equations, Comm. Pure Appl. Math. 41 (1988) 891 - 907.
21. S. Kawashima and R. Kobayashi, Decay estimates and large time behavior of solutions to the drift-diffusion system, preprint, Kyushu University.
22. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970) 399 - 415.
23. n/2 initial data and its application to the blow-up rate, preprint, Tohoku University, Tsuda University.
24. M. Kurokiba, T. Nagai and T. Ogawa, The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system, Commun. Pure Appl. Anal. 5 (2006) 97 - 106.
25. M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Diff. Integral Eqns. 16 (2003) 427 - 452.
26. p arising from the semiconductor device simulation, J. Math. Anal. Appl. 342 (2008) 1052 - 1067.
27. V. A. Liskevich and Y. A. Semenov, Some problems on Markov semigroups, in Schrodinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, 163 - 217, Math. Top. 11 (Akademie Verlag, 1996), pp. 163 - 217.
28. H. Miura, Remark on uniqueness of mild solutions to the Navier - Stokes equations, J. Funct. Anal. 218 (2005) 110 - 129.
29. M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl. 49 (1975) 215 - 225.
30. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995) 581 - 601.
31. T. Nagai, Global existence and blow-up of solutions to a chemotaxis system, Nonlinear Anal. T.M.A. 47 (2001) 777 - 787.
32. T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger - Moser inequality to a parabolic system of chemotaxis, Funk. Ekvac. 40 (1997) 411 - 433.
33. n, Funk. Ekvac. 46 (2003) 383-407.
34. T. Nagai and T. Yamada, Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space, J. Math. Anal. Appl. 336 (2007) 704 - 726.
35. K. Nishihara, Asymptotic profile of solutions to a parabolic system of chemotaxis in one dimensional space, preprint, Waseda University.
36. q decay estimates of solutions to the drift-diffusion system, Master Thesis, Kyushu University, 2004.
37. T. Senba and T. Suzuki, Behavior of solutions to a system related to chemotaxis, Nonlinear Anal. 47 (2001) 2551 - 2562.
38. Y. Shibata and S. Shimizu, A decay property of the Fourier transform and its application to the Stokes problem, J. Math. Fluid Mech. 3 (2001) 213 - 230.
39. A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Jpn. 45 (1997) 241 - 265.
40. T. Yamada, Higher-order asymptotic expansions for a parabolic system modeling chemotaxis in the whole space, preprint, Hiroshima University.
41. W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math., Vol. 120 (Springer-Verlag, 1989).