Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis-haptotaxis

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain Omega subset of R-N (N = 1, 2): {p(t) = Delta p - del . p (alpha/1 + c del c + rho del w) + lambda p(1 - p), x is an element of Omega, t > 0, c(t) = Delta c - c - mu pc, x is an element of Omega, t > 0, w(t )- gamma p(1-w),( )x is an element of Omega, t > 0, where alpha, rho, lambda, mu and gamma are positive parameters. For any reasonably regular initial data (P-0, c(0), w(0)), we prove the global boundedness (L-infinity-norm) of p via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution (p, c, w) converges to (1, 0, 1) with an explicit exponential rate as time tends to infinity.