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

Abstract

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain $\Omega \subset {\mathbb{R}}^N$ ($N=1,2$): $$\label{0} \left\{\begin{array}{ll} p_t=Δp-\nabla\cdotp p(\displaystyle\frac α{1+c}\nabla c+ρ\nabla w)+λp(1-p),\,& x\in Ω, t>0, c_t=Δc-c-μpc,\, &x\in Ω, t>0,\\ w_t= γp(1-w),\,& x\in Ω, t>0, \end{array}\right. $$ 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.