American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
  • DCDS-B Home
  • This Issue
  • Next Article
    Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model
December  2013, 18(10): 2537-2568. doi: 10.3934/dcdsb.2013.18.2537

Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$

Sachiko Ishida 1, , Yusuke Maeda 1, and  Tomomi Yokota 1,

1. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, Japan

Received  November 2012 Revised  February 2013 Published  October 2013

This paper gives the gradient estimate for solutions to the quasilinear non-degenerate parabolic-parabolic Keller-Segel system (KS) on the whole space $\mathbb{R}^N$. The gradient estimate for (KS) on bounded domains is known as an application of Amann's existence theory in [1]. However, in the whole space case it seems necessary to derive the gradient estimate directly. The key to the proof is a modified Bernstein's method. The result is useful to obtain the whole space version of the global existence result by Tao-Winkler [13] except for the boundedness.
Keywords: blow-up., Quasilinear non degenerate Keller-Segel systems.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B3.
Citation: Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537
References:
[1]

Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256.  Google Scholar

[2]

Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. xxxvi+335 pp. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[4]

Comm. Partial Differential Equations, 22 (1997), 1647-1669. doi: 10.1080/03605309708821314.  Google Scholar

[5]

J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[6]

J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[7]

Math. Ann., 312 (1998), 319-340. doi: 10.1007/s002080050224.  Google Scholar

[8]

(Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 xi+648 pp.  Google Scholar

[9]

Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[10]

Differential Integral Equations, 19 (2006), 841-876.  Google Scholar

[11]

Differential Integral Equations, 20 (2007), 133-180.  Google Scholar

[12]

J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[13]

J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.  Google Scholar

show all references

References:
[1]

Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256.  Google Scholar

[2]

Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. xxxvi+335 pp. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[4]

Comm. Partial Differential Equations, 22 (1997), 1647-1669. doi: 10.1080/03605309708821314.  Google Scholar

[5]

J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[6]

J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[7]

Math. Ann., 312 (1998), 319-340. doi: 10.1007/s002080050224.  Google Scholar

[8]

(Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 xi+648 pp.  Google Scholar

[9]

Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[10]

Differential Integral Equations, 19 (2006), 841-876.  Google Scholar

[11]

Differential Integral Equations, 20 (2007), 133-180.  Google Scholar

[12]

J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[13]

J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[1]

Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218
[2]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376
[3]

Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099
[4]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032
[5]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
[6]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[7]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194
[8]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011
[9]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060
[10]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[11]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[12]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[13]

Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021045
[14]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[15]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[16]

John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004
[17]

Wen Si. Response solutions for degenerate reversible harmonic oscillators. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3951-3972. doi: 10.3934/dcds.2021023
[18]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026
[19]

Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3465-3488. doi: 10.3934/dcds.2021004
[20]

Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021070

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences