# American Institute of Mathematical Sciences

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

## 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

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.

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. [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.
 [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] 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 [3] 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 [4] 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 [5] 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 [6] 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 - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299 [7] 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 [8] 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 - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43 [9] 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 [10] 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 [11] 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 [12] Juntang Ding, Xuhui Shen. Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-12. doi: 10.3934/dcdsb.2018135 [13] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [14] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [15] Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010 [16] Juan Luis Vázquez. Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 1-35. doi: 10.3934/dcds.2007.19.1 [17] Pablo Álvarez-Caudevilla, V. A. Galaktionov. Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators. Communications on Pure & Applied Analysis, 2016, 15 (1) : 261-286. doi: 10.3934/cpaa.2016.15.261 [18] Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47 [19] Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 [20] Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941

2016 Impact Factor: 0.826

Article outline