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

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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

Abstract
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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 and 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.
[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.
[3]	P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math., 68 (1995), 229-239.
[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.
[5]	P. Biler and T. Nadzieja, Growth and accretion of mass in an astrophysical model. II, Appl. Math. (Warsaw), 23 (1995), 351-361.
[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.
[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.
[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.
[9]	M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations, 4 (1999), 163-196.
[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.
[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.
[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.
[13]	P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[22]	V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105. doi: 10.1016/0022-5193(73)90149-5.
[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.
[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.
[25]	T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.
[26]	T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston, Inc., Boston, MA, 2005.
[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.
[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.

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.
[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.
[3]	P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math., 68 (1995), 229-239.
[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.
[5]	P. Biler and T. Nadzieja, Growth and accretion of mass in an astrophysical model. II, Appl. Math. (Warsaw), 23 (1995), 351-361.
[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.
[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.
[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.
[9]	M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations, 4 (1999), 163-196.
[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.
[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.
[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.
[13]	P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[22]	V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105. doi: 10.1016/0022-5193(73)90149-5.
[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.
[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.
[25]	T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.
[26]	T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston, Inc., Boston, MA, 2005.
[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.
[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.

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