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 Ω ⊂ ℝ^N (N = 1, 2): (Equation presented) where α, ρ, λ, μ and γ are positive parameters. For any reasonably regular initial data (p₀, c₀, w₀), we prove the global boundedness (L^∞-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.