Existence and Asymptotic Behavior of Solutions to a Quasi-linear Hyperbolic-Parabolic Model of Vasculogenesis

We consider a hyperbolic-parabolic model of vasculogenesis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem, using suitable energy estimates. Since this model does not enter in the classical framework of dissipative problems, we analyze it combining the features of the hyperbolic and the parabolic parts. Moreover we study the asymptotic behavior of those solutions showing their decay rates by means of detailed analysis of the Green function for the linearized problem.

[1]  Shuichi Kawashima,et al.  Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics , 1984 .

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

[3]  B. Perthame,et al.  Derivation of hyperbolic models for chemosensitive movement , 2005, Journal of mathematical biology.

[4]  D. Dormann,et al.  Chemotactic cell movement during Dictyostelium development and gastrulation. , 2006, Current opinion in genetics & development.

[5]  R. NATALINI,et al.  Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy , 2007 .

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

[7]  R. Natalini,et al.  On Relaxation Hyperbolic Systems Violating the Shizuta–Kawashima Condition , 2010 .

[8]  B. Perthame,et al.  Kinetic Models for Chemotaxis and their Drift-Diffusion Limits , 2004 .

[9]  F. Bouchut.  Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws , 1999 .

[10]  Luigi Preziosi,et al.  A review of vasculogenesis models , 2005 .

[11]  R. Natalini,et al.  Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis , 2010 .

[12]  Roberto Natalini,et al.  GLOBAL EXISTENCE OF SMOOTH SOLUTIONS FOR PARTIALLYDISSIPATIVE HYPERBOLIC SYSTEMS WITH A CONVEX ENTROPYB , 2002 .

[13]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[14]  J. Segall,et al.  The great escape: when cancer cells hijack the genes for chemotaxis and motility. , 2005, Annual review of cell and developmental biology.

[15]  Angela Stevens,et al.  The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems , 2000, SIAM J. Appl. Math..

[16]  M. D. Francesco,et al.  Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models , 2008, 0807.3852.

[17]  DianqingWU.  Signaling mechanisms for regulation of chemotaxis , 2005 .

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

[19]  L. Segel,et al.  Model for chemotaxis. , 1971, Journal of theoretical biology.

[20]  Shuichi Kawashima,et al.  On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws , 1988 .

[21]  Shuichi Kawashima,et al.  Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation , 1985 .

[22]  L. Preziosi,et al.  On the stability of homogeneous solutions to some aggregation models , 2003 .

[23]  N. Ferrara,et al.  The biology of VEGF and its receptors , 2003, Nature Medicine.

[24]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes , 2000, SIAM J. Appl. Math..

[25]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[26]  Thomas Hillen,et al.  Hyperbolic models for chemotaxis in 1-D , 2000 .

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

[28]  Gui-Qiang G. Chen,et al.  The Cauchy Problem for the Euler Equations for Compressible Fluids , 2002 .

[29]  Wen-An Yong,et al.  Entropy and Global Existence for Hyperbolic Balance Laws , 2004 .

[30]  R. Natalini,et al.  Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis , 2009 .

[31]  R K Jain,et al.  Growth factors: Formation of endothelial cell networks , 2000, Nature.

[32]  P. Carmeliet.  Mechanisms of angiogenesis and arteriogenesis , 2000, Nature Medicine.

[33]  A. Karsan,et al.  Signaling pathways induced by vascular endothelial growth factor (review). , 2000, International journal of molecular medicine.