Collapsing steady states of the Keller–Segel system

We consider the boundary value problem: which is equivalent to the stationary Keller–Segel system from chemotaxis. Here is a smooth and bounded domain. We show that given any two non-negative integers k, l with k + l ≥ 1, for e sufficiently small, there exists a solution ue for which develops asymptotically k interior Dirac deltas with weight 8π and l boundary deltas with weight 4π. Location of blow-up points is characterized explicitly in terms of Green's function of the Neumann problem.

[1]  J. J. L. Velázquez,et al.  Stability of Some Mechanisms of Chemotactic Aggregation , 2002, SIAM J. Appl. Math..

[2]  Juan J. L. Velázquez.  Well-posedness of a model of point dynamics for a limit of the Keller-Segel system , 2004 .

[3]  T. Nagai,et al.  Global existence and blowup of solutions to a chemotaxis system , 2001 .

[4]  Takashi Suzuki,et al.  Chemotactic collapse in a parabolic system of mathematical biology , 2000 .

[5]  Benoît Perthame,et al.  Optimal critical mass in the two dimensional Keller–Segel model in R2 , 2004 .

[6]  Dirk Horstmann,et al.  On the existence of radially symmetric blow-up solutions for the Keller–Segel model , 2002, Journal of mathematical biology.

[7]  Leo P. Kadanoff,et al.  Diffusion, attraction and collapse , 1999 .

[8]  M. A. Herrero,et al.  Singularity patterns in a chemotaxis model , 1996 .

[9]  Changfeng Gui,et al.  On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems , 2000, Canadian Journal of Mathematics.

[10]  W. Jäger,et al.  On explosions of solutions to a system of partial differential equations modelling chemotaxis , 1992 .

[11]  J. J. L. Velázquez,et al.  Point Dynamics in a Singular Limit of the Keller--Segel Model 2: Formation of the Concentration Regions , 2004, SIAM J. Appl. Math..

[12]  Changfeng Gui,et al.  Multiple interior peak solutions for some singularly perturbed neumann problems , 1999 .

[13]  E. N. Dancer,et al.  Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability , 1999, Advances in Differential Equations.

[14]  M. Pino,et al.  Erratum: "Two-bubble solutions in the super-critical Bahri-Coron's problem" [Calc. Var. Partial Differential Equations 16 (2003), no. 2, 113--145; MR1956850] , 2004 .

[15]  Stephen Childress,et al.  Chemotactic Collapse in Two Dimensions , 1984 .

[16]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[17]  Jerome Percus,et al.  Nonlinear aspects of chemotaxis , 1981 .

[18]  M. A. Herrero,et al.  A blow-up mechanism for a chemotaxis model , 1997 .

[19]  P. Esposito,et al.  On the existence of blowing-up solutions for a mean field equation ? ? The first and second authors , 2005 .

[20]  Takashi Suzuki,et al.  Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities , 1990 .

[21]  M. Pino,et al.  Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data , 2005 .

[22]  P. Felmer,et al.  Two-bubble solutions in the super-critical Bahri-Coron's problem , 2003 .

[23]  Takashi Suzuki,et al.  Weak Solutions to a Parabolic-Elliptic System of Chemotaxis , 2002 .

[24]  E. N. Dancer,et al.  MULTIPEAK SOLUTIONS FOR A SINGULARLY PERTURBED NEUMANN PROBLEM , 1999 .

[25]  Self-similar blow-up for a diffusion-attraction problem , 2004, math/0404283.

[26]  Haim Brezis,et al.  Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions , 1991 .

[27]  M. Pino,et al.  Singular limits in Liouville-type equations , 2005 .

[28]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[29]  R. Schaaf.  Stationary solutions of chemotaxis systems , 1985 .

[30]  Benoît Perthame,et al.  Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions , 2004 .

[31]  F. Pacard,et al.  Construction of singular limits for a semilinear elliptic equation in dimension 2 , 1997 .

[32]  M. A. Herrero,et al.  Chemotactic collapse for the Keller-Segel model , 1996, Journal of mathematical biology.

[33]  Juncheng Wei,et al.  Steady State Solutions of a Reaction‐Diffusion System Modeling Chemotaxis , 2002 .

[34]  Juncheng Wei,et al.  Singular Limits of a Two-Dimensional Boundary Value Problem Arising in Corrosion Modelling , 2006 .

[35]  V. Nanjundiah,et al.  Chemotaxis, signal relaying and aggregation morphology. , 1973, Journal of theoretical biology.

[36]  Matthias Winter,et al.  Multiple boundary peak solutions for some singularly perturbed Neumann problems , 2000 .

[37]  Chiun-Chuan Chen,et al.  Sharp estimates for solutions of multi‐bubbles in compact Riemann surfaces , 2002 .

[38]  Juncheng Wei,et al.  Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity Part I: N=3 , 2004 .

[39]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .