On a hyperbolic-parabolic system modeling chemotaxis

Abstract

We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic-parabolic system derived from the Keller-Segel model describing chemotaxis. It is shown that local smooth solution blows up if and only if the accumulation of the L∞ norm of the solution reaches infinity within the lifespan. Our blowup criteria are consistent with the chemotaxis phenomenon that the movement of cells (bacteria) is driven by the gradient of the chemical concentration. Furthermore, we study the long-time dynamics when the initial data is sufficiently close to a constant positive steady state. By using a new Fourier method adapted to the linear flow, it is shown that the smooth solution exists for all time and converges exponentially to the constant steady state with a frequency-dependent decay rate as time goes to infinity. © 2011 World Scientific Publishing Company.