-
Previous Article
Demography in epidemics modelling
- CPAA Home
- This Issue
-
Next Article
Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque
Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion
1. | GREMAQ, CNRS UMR 5604, INRA UMR 1291, Université de Toulouse, 21 Allée de Brienne, F--31000 Toulouse, France |
2. | Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, F–31062 Toulouse cedex 9 |
References:
[1] |
A. Blanchet, E. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, preprint, arXiv:1009.0134. |
[2] |
A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model, Comm. Pure Appl. Math., 61 (2008), 1449-1481. |
[3] |
A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. |
[4] |
A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32 pp. (electronic). |
[5] |
P.-H. Chavanis and C. Sire, Anomalous diffusion and collapse of self-gravitating Langevin particles in $D$ dimensions, Phys. Rev. E, 69 (2004), 016116. |
[6] |
J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $R^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616. |
[7] |
P. L. Felmer, A. Quaas, M. Tang and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 105-119. |
[8] |
R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. J. Math., 2 (1931), 259-288. |
[9] |
M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754. |
[10] |
M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. |
[11] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[12] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. |
[13] |
E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[14] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266. |
[15] |
E. H. Lieb and M. Loss, "Analysis,'' Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, second ed., 2001. |
[16] |
P. M. Lushnikov, Critical chemotactic collapse, Phys. Lett. A, 374 (2010), 1678-1685. |
[17] |
T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc., 37 (2000), 721-733. |
[18] |
Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34. |
[19] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. |
[20] |
J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. |
[21] |
C. Sire and P.-H. Chavanis, Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions, Phys. Rev. E, 66 (2002), 046133. |
[22] |
C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations, Phys. Rev. E, 78 (2008), 061111. |
[23] |
D. Slepčev and M. C. Pugh, Selfsimilar blowup of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. |
[24] |
Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. |
[25] |
Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144. |
[26] |
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. |
[27] |
M. Tang, Uniqueness of positive radial solutions for $\Delta u - u + u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160. |
show all references
References:
[1] |
A. Blanchet, E. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, preprint, arXiv:1009.0134. |
[2] |
A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model, Comm. Pure Appl. Math., 61 (2008), 1449-1481. |
[3] |
A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. |
[4] |
A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32 pp. (electronic). |
[5] |
P.-H. Chavanis and C. Sire, Anomalous diffusion and collapse of self-gravitating Langevin particles in $D$ dimensions, Phys. Rev. E, 69 (2004), 016116. |
[6] |
J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $R^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616. |
[7] |
P. L. Felmer, A. Quaas, M. Tang and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 105-119. |
[8] |
R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. J. Math., 2 (1931), 259-288. |
[9] |
M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754. |
[10] |
M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. |
[11] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[12] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. |
[13] |
E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[14] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266. |
[15] |
E. H. Lieb and M. Loss, "Analysis,'' Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, second ed., 2001. |
[16] |
P. M. Lushnikov, Critical chemotactic collapse, Phys. Lett. A, 374 (2010), 1678-1685. |
[17] |
T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc., 37 (2000), 721-733. |
[18] |
Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34. |
[19] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. |
[20] |
J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. |
[21] |
C. Sire and P.-H. Chavanis, Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions, Phys. Rev. E, 66 (2002), 046133. |
[22] |
C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations, Phys. Rev. E, 78 (2008), 061111. |
[23] |
D. Slepčev and M. C. Pugh, Selfsimilar blowup of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. |
[24] |
Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. |
[25] |
Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144. |
[26] |
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. |
[27] |
M. Tang, Uniqueness of positive radial solutions for $\Delta u - u + u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160. |
[1] |
Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911 |
[2] |
Jacob Bedrossian, Nancy Rodríguez. Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1279-1309. doi: 10.3934/dcdsb.2014.19.1279 |
[3] |
Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897 |
[4] |
Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002 |
[5] |
Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801 |
[6] |
K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure and Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51 |
[7] |
Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817 |
[8] |
Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036 |
[9] |
Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101 |
[10] |
Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks and Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857 |
[11] |
F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91 |
[12] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[13] |
Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703 |
[14] |
Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations and Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003 |
[15] |
Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic and Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012 |
[16] |
Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks and Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181 |
[17] |
Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003 |
[18] |
Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046 |
[19] |
Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471 |
[20] |
Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]