Boundedness of the attraction–repulsion Keller–Segel system

a b s t r a c t This paper considers the initial-boundary value problem of the attraction-repulsion Keller-Segel model describing aggregation of Microglia in the central nervous system in Alzheimer's disease due to the interaction of chemoattractant and chemorepellent. If repulsion dominates over attraction, we show the global existence of classical solution in two dimensions and weak solution in three dimensions with large initial data. u(x, 0) = u0(x) ,τ v (x, 0) = τv 0(x) ,τ w (x, 0) = τw 0(x) ,x ∈ Ω,

[1]  Youshan Tao,et al.  Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity , 2011, 1106.5345.

[2]  Takashi Suzuki,et al.  Global existence and blow-up for a system describing the aggregation of microglia , 2014, Appl. Math. Lett..

[3]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[4]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[5]  Zhi-An Wang,et al.  Classical solutions and steady states of an attraction–repulsion chemotaxis in one dimension , 2012, Journal of biological dynamics.

[6]  Dirk Horstmann,et al.  Blow-up in a chemotaxis model without symmetry assumptions , 2001, European Journal of Applied Mathematics.

[7]  Michael Winkler,et al.  Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model , 2010 .

[8]  P. Laurençot,et al.  Global existence and convergence to steady states in a chemorepulsion system , 2008 .

[9]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[10]  Nicholas D. Alikakos,et al.  LP Bounds of solutions of reaction-diffusion equations , 1979 .

[11]  Zhian Wang.  Mathematics of traveling waves in chemotaxis --Review paper-- , 2012 .

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

[13]  Michael Winkler,et al.  Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system , 2011, 1112.4156.

[14]  Herbert Amann,et al.  Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems , 1993 .

[15]  Yung-Sze Choi,et al.  Prevention of blow-up by fast diffusion in chemotaxis , 2010 .

[16]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[17]  Zuzanna Szymańska,et al.  On the global existence of solutions to an aggregation model , 2008 .

[18]  Ansgar Jüngel,et al.  Cross Diffusion Preventing Blow-Up in the Two-Dimensional Keller-Segel Model , 2011, SIAM J. Math. Anal..

[19]  Youshan Tao,et al.  Competing effects of attraction vs. repulsion in chemotaxis , 2013 .

[20]  Dirk Horstmann,et al.  Boundedness vs. blow-up in a chemotaxis system , 2005 .

[21]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[22]  Leah Edelstein-Keshet,et al.  Chemotactic signaling, microglia, and Alzheimer’s disease senile plaques: Is there a connection? , 2003, Bulletin of mathematical biology.

[23]  Jia Liu.  Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension , 2011 .

[24]  B. Perthame,et al.  Travelling plateaus for a hyperbolic Keller–Segel system with attraction and repulsion: existence and branching instabilities , 2010, 1009.6090.

[25]  Toshitaka Nagai,et al.  Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains , 2001 .

[26]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[27]  Zhian Wang,et al.  Asymptotic dynamics of the one‐dimensional attraction–repulsion Keller–Segel model , 2015 .

[28]  Ping Liu,et al.  Pattern Formation of the Attraction-Repulsion Keller-Segel System , 2013 .