Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure

In this article, we study a one-dimensional hyperbolic quasi-linear model of chemotaxis with a non-linear pressure and we consider its stationary solutions, in particular with vacuum regions. We study both cases of the system set on the whole line $\Er$ and on a bounded interval with no-flux boundary conditions. In the case of the whole line $\Er$, we find only one stationary solution, up to a translation, formed by a positive density region (called bump) surrounded by two regions of vacuum. However, in the case of a bounded interval, an infinite of stationary solutions exists, where the number of bumps is limited by the length of the interval. We are able to compare the value of an energy of the system for these stationary solutions. Finally, we study the stability of these stationary solutions through numerical simulations.

[1]  Vincent Calvez,et al.  Blow-up, Concentration Phenomenon and Global Existence for the Keller–Segel Model in High Dimension , 2010, 1003.4182.

[2]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[3]  L. Preziosi,et al.  Modeling the early stages of vascular network assembly , 2003, The EMBO journal.

[4]  Mazen Saad,et al.  Journal of Computational and Applied Mathematics Finite Volume Methods for Degenerate Chemotaxis Model , 2022 .

[5]  Non-Linear Stability of Gaseous Stars , 2002, math-ph/0210045.

[6]  Vincent Calvez,et al.  The parabolic-parabolic Keller-Segel model in R2 , 2008 .

[7]  R. Kowalczyk,et al.  Preventing blow-up in a chemotaxis model , 2005 .

[8]  Roberto Natalini,et al.  A well-balanced numerical scheme for a one dimensional quasilinear hyperbolic model of chemotaxis , 2012, 1211.4010.

[9]  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 .

[10]  J. Carrillo,et al.  A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure , 2014, 1402.4252.

[11]  Francis Filbet,et al.  Approximation of Hyperbolic Models for Chemosensitive Movement , 2005, SIAM J. Sci. Comput..

[12]  Tai-Ping Liu,et al.  Solutions of Euler-Poisson Equations¶for Gaseous Stars , 2002 .

[13]  Martin Burger,et al.  Stationary States and Asymptotic Behavior of Aggregation Models with Nonlinear Local Repulsion , 2013, SIAM J. Appl. Dyn. Syst..

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

[15]  José A. Carrillo,et al.  Volume effects in the Keller-Segel model : energy estimates preventing blow-up , 2006 .

[16]  N. Masmoudi,et al.  Well and ill-posedness for compressible Euler equations with vacuum , 2012 .

[17]  Cristiana Di Russo,et al.  Existence and Asymptotic Behavior of Solutions to a Quasi-linear Hyperbolic-Parabolic Model of Vasculogenesis , 2011, SIAM J. Math. Anal..

[18]  Roberto Natalini,et al.  A Numerical Comparison Between Degenerate Parabolic and Quasilinear Hyperbolic Models of Cell Movements Under Chemotaxis , 2014, J. Sci. Comput..

[19]  B. Perthame.  Transport Equations in Biology , 2006 .

[20]  Matteo Semplice,et al.  3D simulations of early blood vessel formation , 2007, J. Comput. Phys..

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

[22]  Thomas Hillen,et al.  Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding , 2001, Adv. Appl. Math..

[23]  Tetsuya Yamada,et al.  Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space , 2007 .

[24]  Cristiana Di Russo,et al.  Analysis and Numerical Approximations of Hydrodynamical Models of Biological Movements , 2012 .

[25]  L Preziosi,et al.  Percolation, morphogenesis, and burgers dynamics in blood vessels formation. , 2003, Physical review letters.

[26]  Zhen Lei,et al.  Well-posedness of 1-D compressible Euler-Poisson equations with physical vacuum , 2011, 1105.0529.