Global Dynamics of a Hyperbolic-Parabolic Model Arising from Chemotaxis

We prove global existence and qualitative behavior of classical solutions for a hyperbolic-parabolic system describing chemotaxis on bounded domains. It is shown that classical solutions to the initial-boundary value problem of the one-dimensional model exist globally in time for large initial data, and the solutions converge to constant equilibrium states exponentially in time, which rigorously demonstrates the collapsing of cell populations in chemotaxis. Moreover, similar results are established for the multidimensional model when the initial data are small.