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

## TL;DR

This paper analyzes the long-term behavior of solutions to a tumor angiogenesis model with chemotaxis and haptotaxis, proving global boundedness and convergence results, especially in one dimension, with explicit rates.

## Contribution

It establishes the global boundedness of solutions and improves existing results on their asymptotic behavior, including explicit convergence rates in one dimension.

## Key findings

- Solutions are globally bounded in $L^
abla$-norm.
- In one dimension, solutions converge exponentially to a steady state.
- The results extend and refine previous asymptotic analyses.

## Abstract

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain $\Omega \subset \mathbb{R}^N$ ($N=1,2$): $$\label{0} \left\{\begin{array}{ll} p_t=\Delta p-\nabla\cdotp p(\displaystyle\frac \alpha {1+c}\nabla c+\rho\nabla w)+\lambda p(1-p),\,& x\in \Omega, t>0, c_t=\Delta c-c-\mu pc,\, &x\in \Omega, t>0,\\ w_t= \gamma p(1-w),\,& x\in \Omega, t>0, \end{array}\right. $$ 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.