Global dynamics in a chemotaxis model describing tumor angiogenesis with/without mitosis in any dimensions

In this work, we study the Neumann initial boundary value problem for a three-component chemotaxis model in any dimensional bounded and smooth domains; this model is used to describe the branching of capillary sprouts during angiogenesis. First, we find three qualitatively simple sufficient conditions for qualitative global boundedness, and then, we establish two types of global stability for bounded solutions in qualitative ways. As a consequence of our findings, the underlying system without chemotaxis and the effect of ECs mitosis can not give rise to pattern formations. Our findings quantify and extend significantly previous studies, which are set in lower dimensional convex domains and are with no qualitative information.Comment: 43 pages, under review in a journa

Global existence of solutions to the chemotaxis system with logistic source under nonlinear Neumann boundary condition

We consider classical solutions to the chemotaxis system with logistic source $f(u) := au-\mu u^2$ under nonlinear Neumann boundary condition $\frac{\partial u}{ \partial \nu } = |u|^{p}$ with $p>1$ in a smooth convex bounded domain $\Omega \subset \mathbb{R}^n$ where $n \geq 2$. This paper aims to show that if $p0$, $n=2$, or $\mu$ is sufficiently large when $n\geq 3$, then the parabolic-elliptic chemotaxis system admits a unique positive global-in-time classical solution that is bounded in $\Omega \times (0, \infty)$. The similar result is also true if $p<\frac{3}{2}$, $n=2$, and $\mu>0$ or $p<\frac{7}{5}$, $n=3$, and $\mu$ is sufficiently large for the parabolic-parabolic chemotaxis system

Analysis of Reaction-Diffusion Models with the Taxis Mechanism

This open access book deals with a rich variety of taxis-type cross-diffusive equations. Particularly, it intends to show the key role played by quasi-energy inequality in the derivation of some necessary a priori estimates. This book addresses applied mathematics and all researchers interested in mathematical development of reaction-diffusion theory and its application and can be a basis for a graduate course in applied mathematics

Analysis of Reaction-Diffusion Models with the Taxis Mechanism

This open access book deals with a rich variety of taxis-type cross-diffusive equations. Particularly, it intends to show the key role played by quasi-energy inequality in the derivation of some necessary a priori estimates. This book addresses applied mathematics and all researchers interested in mathematical development of reaction-diffusion theory and its application and can be a basis for a graduate course in applied mathematics

走性に関する生物行動の数理モデルとその解析

関西学院大