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October  2012, 32(10): 3691-3713. doi: 10.3934/dcds.2012.32.3691

Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains

Yūki Naito 1, and  Takasi Senba 2,

1. 

Department of Mathematics, Faculty of Sciences, Ehime University, Matsuyama, 790-8577
2. 

Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550, Japan

Received  May 2011 Revised  March 2012 Published  May 2012

We consider a parabolic-elliptic system of equations that arises in modelling the chemotaxis in bacteria and the evolution of self-attracting clusters. In the case space dimension $3 \leq N \leq 9$, we will derive criteria of the blow-up rate of solutions, and identify an explicit class of initial data for which the blow-up is of self-similar rate. Our argument is based on the study of the asymptotic properties of backward self-similar solutions to the system together with the intersection comparison principle.
Keywords: Parabolic-elliptic system, blow-up rate, intersection comparison., self-similar solution.
Mathematics Subject Classification: Primary: 35K55, 35B40; Secondary: 34D0.
Citation: Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691
References:
[1]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  Google Scholar

[2]

J. Bebernes and D. Eberly, A description of self-similar blow-up for dimensions $n \geq 3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 1-21.  Google Scholar

[3]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math., 68 (1995), 229-239.  Google Scholar

[4]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math., 66 (1994), 319-334.  Google Scholar

[5]

P. Biler and T. Nadzieja, Growth and accretion of mass in an astrophysical model. II, Appl. Math. (Warsaw), 23 (1995), 351-361.  Google Scholar

[6]

P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model gravitational of particles. II, Colloq. Math., 67 (1994), 297-308.  Google Scholar

[7]

M. P. Brenner, P. Constantin, L. P. Kadanoff, A. Schenkel and S. C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098. doi: 10.1088/0951-7715/12/4/320.  Google Scholar

[8]

X.-Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball, J. Reine Angew. Math., 472 (1996), 17-51.  Google Scholar

[9]

M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations, 4 (1999), 163-196.  Google Scholar

[10]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[11]

Y. Giga, N. Mizoguchi and T. Senba, Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type, Arch. Rational Mech. Anal., 201 (2011), 549-573. doi: 10.1007/s00205-010-0394-7.  Google Scholar

[12]

I. A. Guerra and M. A. Peletier, Self-similar blow-up for a diffusion-attraction problem, Nonlinearity, 17 (2004), 2137-2162. doi: 10.1088/0951-7715/17/6/007.  Google Scholar

[13]

P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[14]

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. doi: 10.1088/0951-7715/10/6/016.  Google Scholar

[15]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Self-similar blowup for a reaction-diffusion system, Journal of Computational and Applied Mathematics, 97 (1998), 99-119. doi: 10.1016/S0377-0427(98)00104-6.  Google Scholar

[16]

M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. doi: 10.1007/BF01445268.  Google Scholar

[17]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194. doi: 10.1007/s002850050049.  Google Scholar

[18]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[19]

J. Matos, Convergence of blow-up solutions of nonlinear heat equations in the supercritical case, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1197-1227. doi: 10.1017/S0308210500019351.  Google Scholar

[20]

N. Mizoguchi and T. Senba, A sufficient condition for type I blowup in a parabolic-elliptic system, J. Differential Equations, 250 (2011), 182-203. doi: 10.1016/j.jde.2010.10.016.  Google Scholar

[21]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[22]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105. doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[23]

T. Senba, Blowup behavior of radial solutions Jäger-Luckhaus system in high dimensional domains, Funkcial Ekvac., 48 (2005), 247-271. doi: 10.1619/fesi.48.247.  Google Scholar

[24]

T. Senba, Type II blowup of solutions to a simplified Keller-Segel system in two dimensional domains, Nonlinear Anal., 66 (2007), 1817-1839. doi: 10.1016/j.na.2006.02.027.  Google Scholar

[25]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.  Google Scholar

[26]

T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston, Inc., Boston, MA, 2005.  Google Scholar

[27]

G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal., 119 (1992), 355-391. doi: 10.1007/BF01837114.  Google Scholar

[28]

G. Wolansky, On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity, J. Anal. Math., 59 (1992), 251-272. doi: 10.1007/BF02790230.  Google Scholar

show all references

References:
[1]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  Google Scholar

[2]

J. Bebernes and D. Eberly, A description of self-similar blow-up for dimensions $n \geq 3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 1-21.  Google Scholar

[3]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math., 68 (1995), 229-239.  Google Scholar

[4]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math., 66 (1994), 319-334.  Google Scholar

[5]

P. Biler and T. Nadzieja, Growth and accretion of mass in an astrophysical model. II, Appl. Math. (Warsaw), 23 (1995), 351-361.  Google Scholar

[6]

P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model gravitational of particles. II, Colloq. Math., 67 (1994), 297-308.  Google Scholar

[7]

M. P. Brenner, P. Constantin, L. P. Kadanoff, A. Schenkel and S. C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098. doi: 10.1088/0951-7715/12/4/320.  Google Scholar

[8]

X.-Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball, J. Reine Angew. Math., 472 (1996), 17-51.  Google Scholar

[9]

M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations, 4 (1999), 163-196.  Google Scholar

[10]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[11]

Y. Giga, N. Mizoguchi and T. Senba, Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type, Arch. Rational Mech. Anal., 201 (2011), 549-573. doi: 10.1007/s00205-010-0394-7.  Google Scholar

[12]

I. A. Guerra and M. A. Peletier, Self-similar blow-up for a diffusion-attraction problem, Nonlinearity, 17 (2004), 2137-2162. doi: 10.1088/0951-7715/17/6/007.  Google Scholar

[13]

P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[14]

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. doi: 10.1088/0951-7715/10/6/016.  Google Scholar

[15]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Self-similar blowup for a reaction-diffusion system, Journal of Computational and Applied Mathematics, 97 (1998), 99-119. doi: 10.1016/S0377-0427(98)00104-6.  Google Scholar

[16]

M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. doi: 10.1007/BF01445268.  Google Scholar

[17]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194. doi: 10.1007/s002850050049.  Google Scholar

[18]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[19]

J. Matos, Convergence of blow-up solutions of nonlinear heat equations in the supercritical case, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1197-1227. doi: 10.1017/S0308210500019351.  Google Scholar

[20]

N. Mizoguchi and T. Senba, A sufficient condition for type I blowup in a parabolic-elliptic system, J. Differential Equations, 250 (2011), 182-203. doi: 10.1016/j.jde.2010.10.016.  Google Scholar

[21]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[22]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105. doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[23]

T. Senba, Blowup behavior of radial solutions Jäger-Luckhaus system in high dimensional domains, Funkcial Ekvac., 48 (2005), 247-271. doi: 10.1619/fesi.48.247.  Google Scholar

[24]

T. Senba, Type II blowup of solutions to a simplified Keller-Segel system in two dimensional domains, Nonlinear Anal., 66 (2007), 1817-1839. doi: 10.1016/j.na.2006.02.027.  Google Scholar

[25]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.  Google Scholar

[26]

T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston, Inc., Boston, MA, 2005.  Google Scholar

[27]

G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal., 119 (1992), 355-391. doi: 10.1007/BF01837114.  Google Scholar

[28]

G. Wolansky, On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity, J. Anal. Math., 59 (1992), 251-272. doi: 10.1007/BF02790230.  Google Scholar

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