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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

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