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

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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.
    Mathematics Subject Classification: Primary: 35K55, 35B40; Secondary: 34D05.


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  • [1]

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


    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.


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


    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.


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


    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.


    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.


    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.


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


    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.


    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.


    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.


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


    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.


    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.


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


    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.


    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.


    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.


    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.


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


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


    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.


    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.


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


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


    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.


    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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