TY - JOUR

T1 - Finite time blow up for a solution to system of the drift–diffusion equations in higher dimensions

AU - Kurokiba, Masaki

AU - Ogawa, Takayoshi

N1 - Funding Information:
The work of the first author is partially supported by JSPS Grant-in-aid for Scientific Research C # 24540201. The work of the second author is partially supported by JSPS Grant-in-aid for Scientific Research S #25220702.
Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - We discuss the existence of a blow-up solution for a multi-component parabolic–elliptic drift–diffusion model in higher space dimensions. We show that the local existence, uniqueness and well-posedness of a solution in the weighted L2 spaces. Moreover we prove that if the initial data satisfies certain conditions, then the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift–diffusion equation proved by Nagai (J Inequal Appl 6:37–55, 2001) and Nagai et al. (Hiroshima J Math 30:463–497, 2000) and gravitational interaction of particles by Biler (Colloq Math 68:229–239, 1995), Biler and Nadzieja (Colloq Math 66:319–334, 1994, Adv Differ Equ 3:177–197, 1998). We generalize the result in Kurokiba and Ogawa (Differ Integral Equ 16:427–452, 2003, Differ Integral Equ 28:441–472, 2015) and Kurokiba (Differ Integral Equ 27(5–6):425–446, 2014) for the multi-component problem and give a sufficient condition for the finite time blow up of the solution. The condition is different from the one obtained by Corrias et al. (Milan J Math 72:1–28, 2004).

AB - We discuss the existence of a blow-up solution for a multi-component parabolic–elliptic drift–diffusion model in higher space dimensions. We show that the local existence, uniqueness and well-posedness of a solution in the weighted L2 spaces. Moreover we prove that if the initial data satisfies certain conditions, then the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift–diffusion equation proved by Nagai (J Inequal Appl 6:37–55, 2001) and Nagai et al. (Hiroshima J Math 30:463–497, 2000) and gravitational interaction of particles by Biler (Colloq Math 68:229–239, 1995), Biler and Nadzieja (Colloq Math 66:319–334, 1994, Adv Differ Equ 3:177–197, 1998). We generalize the result in Kurokiba and Ogawa (Differ Integral Equ 16:427–452, 2003, Differ Integral Equ 28:441–472, 2015) and Kurokiba (Differ Integral Equ 27(5–6):425–446, 2014) for the multi-component problem and give a sufficient condition for the finite time blow up of the solution. The condition is different from the one obtained by Corrias et al. (Milan J Math 72:1–28, 2004).

KW - 35K55

KW - 35Q60

KW - Primary 35K15

KW - Secondary 78A35

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U2 - 10.1007/s00209-016-1654-5

DO - 10.1007/s00209-016-1654-5

M3 - Article

AN - SCOPUS:84968608629

SN - 0025-5874

VL - 284

SP - 231

EP - 253

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -