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Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems
Stability and instability of solutions to the drift-diffusion system
| 1. | Tohoku University, Mathematical Institute, Sendai 980-8578, Japan |
| 2. | Mathematical Institute, Tohoku University, Sendai 980-8578, Japan |
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.
References:
| [1] |
P. Biler,
Existence and nonexistence of solutions for a model of gravitational interaction of particles, Ⅲ, Colloq. Math., 68 (1995), 229-239.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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.
|
| [12] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
| [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.
|
| [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. |
| [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. |
| [16] |
T. Ogawa,
Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type, Differential Integral Equations, 21 (2008), 1113-1154.
|
| [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. |
| [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.
|
| [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.
|
| [20] |
H. Wakui, Asymptotic behavior of a weak solution to a degenerate drift-diffusion equation, Master course thesis, Tohoku University, 2012. |
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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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.
|
| [12] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
| [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.
|
| [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. |
| [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. |
| [16] |
T. Ogawa,
Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type, Differential Integral Equations, 21 (2008), 1113-1154.
|
| [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. |
| [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.
|
| [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.
|
| [20] |
H. Wakui, Asymptotic behavior of a weak solution to a degenerate drift-diffusion equation, Master course thesis, Tohoku University, 2012. |
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