American Institute of Mathematical Sciences

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

 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.
Citation: Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 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. 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. Google Scholar [3] P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III,, Colloq. Math., 68 (1995), 229. 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. Google Scholar [5] P. Biler and T. Nadzieja, Growth and accretion of mass in an astrophysical model. II,, Appl. Math. (Warsaw), 23 (1995), 351. 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. 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. 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. Google Scholar [9] M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation,, Adv. Differential Equations, 4 (1999), 163. Google Scholar [10] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425. 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. 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. doi: 10.1088/0951-7715/17/6/007. Google Scholar [13] P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [22] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theor. Biol., 42 (1973), 63. 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. 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. 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. Google Scholar [26] T. Suzuki, "Free Energy and Self-Interacting Particles,", Progress in Nonlinear Differential Equations and their Applications, 62 (2005). Google Scholar [27] G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations,, Arch. Rational Mech. Anal., 119 (1992), 355. 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. doi: 10.1007/BF02790230. Google Scholar

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References:
 [1] S. Angenent, The zero set of a solution of a parabolic equation,, J. Reine Angew. Math., 390 (1988), 79. 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. Google Scholar [3] P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III,, Colloq. Math., 68 (1995), 229. 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. Google Scholar [5] P. Biler and T. Nadzieja, Growth and accretion of mass in an astrophysical model. II,, Appl. Math. (Warsaw), 23 (1995), 351. 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. 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. 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. Google Scholar [9] M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation,, Adv. Differential Equations, 4 (1999), 163. Google Scholar [10] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425. 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. 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. doi: 10.1088/0951-7715/17/6/007. Google Scholar [13] P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [22] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theor. Biol., 42 (1973), 63. 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. 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. 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. Google Scholar [26] T. Suzuki, "Free Energy and Self-Interacting Particles,", Progress in Nonlinear Differential Equations and their Applications, 62 (2005). Google Scholar [27] G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations,, Arch. Rational Mech. Anal., 119 (1992), 355. 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. doi: 10.1007/BF02790230. Google Scholar
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