American Institute of Mathematical Sciences

Advanced
December  2017, 6(4): 587-597. doi: 10.3934/eect.2017029

Stability and instability of solutions to the drift-diffusion system

Takayoshi Ogawa 1, and  Hiroshi Wakui 2,

1. 

Tohoku University, Mathematical Institute, Sendai 980-8578, Japan
2. 

Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Received  March 2017 Revised  August 2017 Published  September 2017

Fund Project: The work of T. Ogawa and H. Wakui are partially supported by grant in aid for Scientific Research S #24220702 of JSPS.

We consider the large time behavior of a solution to a drift-diffusion equation for degenerate and non-degenerate type. We show an instability and uniform unbounded estimate for the semi-linear case and uniform bound and convergence to the stationary solution for the case of mass critical degenerate case for higher space of dimension bigger than two.

Keywords: Drift diffusion system, stability, boundedness, finite time blow up, scaling critical.
Mathematics Subject Classification: Primary: 35K15, 35K55, 35Q60; Secondary: 78A35.
Citation: Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the drift-diffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587-597. doi: 10.3934/eect.2017029
References:
[1]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles, Ⅲ, Colloq. Math., 68 (1995), 229-239.   Google Scholar

[2]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interactions of particles Ⅰ, Colloq. Math., 66 (1994), 319-334.   Google Scholar

[3]

A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

[4]

V. Calvez, L. Corrias and M. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. Partial Differential Equations, 37 (2012), 561-584.  doi: 10.1080/03605302.2012.655824.  Google Scholar

[5]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis system in hight space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[6]

E. Feireisl and P. Laurençot, Non-isothermal Smoluchowski-Poisson equations as a singular limit of the Navier-Stokes-Fourier-Poisson system, J. Math. Pures Appl., 88 (2007), 325-349.  doi: 10.1016/j.matpur.2007.07.002.  Google Scholar

[7]

A. Kimijima, K. Nakagawa and T. Ogawa, Threshold of global behavior of solutions to a degenerate drift-diffusion system in between two critical exponents, Calc. Var. Partial Differential Equations, 53 (2015), 441-472.  doi: 10.1007/s00526-014-0755-4.  Google Scholar

[8]

T. Kobayashi and T. Ogawa, Fluid mechanical approximation to the degenerated drift-diffusion and chemotaxis equations in barotropic model, Indiana Univ. Math. J., 62 (2013), 1021-1054.  doi: 10.1512/iumj.2013.62.5017.  Google Scholar

[9]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations, 16 (2003), 427-452.   Google Scholar

[10]

M. Kurokiba and T. Ogawa, Finite time blow up for a solution to system of the drift-diffusion equations in higher dimensions, Math. Z., 284 (2016), 231-253.  doi: 10.1007/s00209-016-1654-5.  Google Scholar

[11]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar

[12]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[13]

T. Nagai, Blow up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[14]

T. Nagai and T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in $\mathbb{R}^2$, Funkcial. Ekvac., 59 (2016), 67-112.  doi: 10.1619/fesi.59.67.  Google Scholar

[15]

T. Ogawa, Decay and asymptotic behavior of a solution of the Keller-Segel system of degenerate and nondegenerate type, Banach Center Publ., 74 (2006), 161-184.  doi: 10.4064/bc74-0-10.  Google Scholar

[16]

T. Ogawa, Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type, Differential Integral Equations, 21 (2008), 1113-1154.   Google Scholar

[17]

T. Ogawa and H. Wakui, Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions, Anal. Appl. (Singap.), 14 (2016), 145-183.  doi: 10.1142/S0219530515400060.  Google Scholar

[18]

T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, Ⅰ, Generation of the weak solution, Adv. Differential Equations, 14 (2009), 433-476.   Google Scholar

[19]

T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, Ⅱ, Blow-up threshold, Differential Integral Equations, 22 (2009), 1153-1172.   Google Scholar

[20]

H. Wakui, Asymptotic behavior of a weak solution to a degenerate drift-diffusion equation, Master course thesis, Tohoku University, 2012. Google Scholar

show all references

References:
[1]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles, Ⅲ, Colloq. Math., 68 (1995), 229-239.   Google Scholar

[2]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interactions of particles Ⅰ, Colloq. Math., 66 (1994), 319-334.   Google Scholar

[3]

A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

[4]

V. Calvez, L. Corrias and M. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. Partial Differential Equations, 37 (2012), 561-584.  doi: 10.1080/03605302.2012.655824.  Google Scholar

[5]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis system in hight space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[6]

E. Feireisl and P. Laurençot, Non-isothermal Smoluchowski-Poisson equations as a singular limit of the Navier-Stokes-Fourier-Poisson system, J. Math. Pures Appl., 88 (2007), 325-349.  doi: 10.1016/j.matpur.2007.07.002.  Google Scholar

[7]

A. Kimijima, K. Nakagawa and T. Ogawa, Threshold of global behavior of solutions to a degenerate drift-diffusion system in between two critical exponents, Calc. Var. Partial Differential Equations, 53 (2015), 441-472.  doi: 10.1007/s00526-014-0755-4.  Google Scholar

[8]

T. Kobayashi and T. Ogawa, Fluid mechanical approximation to the degenerated drift-diffusion and chemotaxis equations in barotropic model, Indiana Univ. Math. J., 62 (2013), 1021-1054.  doi: 10.1512/iumj.2013.62.5017.  Google Scholar

[9]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations, 16 (2003), 427-452.   Google Scholar

[10]

M. Kurokiba and T. Ogawa, Finite time blow up for a solution to system of the drift-diffusion equations in higher dimensions, Math. Z., 284 (2016), 231-253.  doi: 10.1007/s00209-016-1654-5.  Google Scholar

[11]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar

[12]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[13]

T. Nagai, Blow up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[14]

T. Nagai and T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in $\mathbb{R}^2$, Funkcial. Ekvac., 59 (2016), 67-112.  doi: 10.1619/fesi.59.67.  Google Scholar

[15]

T. Ogawa, Decay and asymptotic behavior of a solution of the Keller-Segel system of degenerate and nondegenerate type, Banach Center Publ., 74 (2006), 161-184.  doi: 10.4064/bc74-0-10.  Google Scholar

[16]

T. Ogawa, Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type, Differential Integral Equations, 21 (2008), 1113-1154.   Google Scholar

[17]

T. Ogawa and H. Wakui, Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions, Anal. Appl. (Singap.), 14 (2016), 145-183.  doi: 10.1142/S0219530515400060.  Google Scholar

[18]

T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, Ⅰ, Generation of the weak solution, Adv. Differential Equations, 14 (2009), 433-476.   Google Scholar

[19]

T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, Ⅱ, Blow-up threshold, Differential Integral Equations, 22 (2009), 1153-1172.   Google Scholar

[20]

H. Wakui, Asymptotic behavior of a weak solution to a degenerate drift-diffusion equation, Master course thesis, Tohoku University, 2012. Google Scholar

[1]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[2]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006
[3]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[4]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875
[5]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052
[6]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558
[7]

Jihoon Lee. Scaling invariant blow-up criteria for simplified versions of Ericksen-Leslie system. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 381-388. doi: 10.3934/dcdss.2015.8.381
[8]

Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure & Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97
[9]

José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[10]

Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[11]

Youshan Tao, Michael Winkler. Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3165-3183. doi: 10.3934/dcdsb.2015.20.3165
[12]

Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182
[13]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
[14]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[15]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[16]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[17]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[18]

Juntang Ding, Xuhui Shen. Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4243-4254. doi: 10.3934/dcdsb.2018135
[19]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021052
[20]

Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021151

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (140)
  • HTML views (213)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences