Wellposedness of solution for an $N$-D chemotaxis-convection model during tumor angiogenesis

TL;DR

This paper proves the global existence and boundedness of solutions for a complex chemotaxis-convection system modeling tumor angiogenesis across various spatial dimensions and parameter conditions.

Contribution

It establishes well-posedness results for a multi-dimensional chemotaxis-convection model with minimal restrictions on parameters.

Findings

Global classical solutions exist under specified conditions.

Solutions remain bounded for all time.

Results apply to various dimensions and parameter regimes.

Abstract

In this paper, we consider the following parabolic-parabolic-elliptic system } \begin{align*} \left\{\aligned & u_t=\Delta u-\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+au-\mu u^{\alpha}, && x\in\Omega, t>0,\\ & v_t=\Delta v+\nabla\cdot(v\nabla w)-v+u,&& x\in\Omega, t>0,\\ & 0=\Delta w-w+u,&& x\in\Omega, t>0\\ \endaligned\right. \end{align*} on a bounded domain $\Omega\subset \mathbb{R}^{N}$ ($N\geq1$) with smooth boundary $\partial \Omega$, where $\mu$, $a$, $\alpha$ are positive constants and $\xi\in\mathbb{R}$. If one of the following cases holds:\\ (i) $N\geq4$ and $\alpha>\frac{4N-4+N\sqrt{2N^2-6N+8}}{2N}$;\\ (ii) $N=3$, $\alpha>2$, for any $\mu>0$ or $\alpha=2$, the index $\mu$ should be suitably big;\\ (iii) $N=2$, $\alpha\geq2$, for any $\mu>0$.\\ Without any restriction on the index $\xi$, for any given suitably regular initial data, the corresponding Neumann initial-boundary problem admits a unique global and bounded classical solution.