Existence and asymptotic behavior of solutions to a quasi-linear hyperbolic-parabolic model of vasculogenesis (Q2840371)

The authors study the global existence of smooth solutions to the Cauchy problem for a hyperbolic-parabolic model of vasculogenesis in multi-dimensions which consists of the compressible damped isentropic Euler equations coupled with a diffusion equation for the concentration of the chemoattractant. This model does not enter in the classical framework of dissipative problems (i.e., it does not satisfy Kawashima-Shizuta's conditions due to the presence of the source terms), the authors combine the features of the hyperbolic and the parabolic parts, and use suitable energy estimates to prove the global existence and uniqueness of smooth solutions provided that initial data are sufficiently small. Moreover, the asymptotic behavior of the global solution showing its decay rates is established by means of detailed analysis of the Green function for the linearized problem.