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 $Omega subset mathbb{R}^N$ ($N=1,2$): $$label{0} left{begin{array}{ll} p_t=Delta p- ablacdotp p(displaystylefrac alpha {1+c} abla c+rho abla w)+lambda p(1-p),,& xin Omega, t>0, c_t=Delta c-c-mu pc,, &xin Omega, t>0, w_t= gamma p(1-w),,& xin Omega, t>0, end{array}right. $$ 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 ($L^infty$-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.