Tian Xiang — Page 2

arXiv2016

Boundedness and exponential convergence of a chemotaxis model for tumor invasion

We revisit the following chemotaxis system modeling tumor invasion \begin{equation*} \begin{cases} u_t=Δu-\nabla \cdot(u\nabla v),& x\inΩ, t>0,\\ v_t=Δv+wz,& x\inΩ, t>0,\\ w_t=-wz,& x\inΩ, t>0,\\ z_t=Δz-z+u, & x\inΩ, t>0,\\ \end{cases} \end{equation*} in a smooth bounded domain $Ω\subset \mathbb{R}^n(n\geq 1)$ with homogeneous Neumann boundary and initial conditions. This model was recently proposed by Fujie et al. \cite{FIY14} as a model for tumor invasion with the role of extracellular matrix incorporated, and was analyzed by Fujie et al. \cite{FIWY16}, showing the uniform boundedness and convergence for $n\leq 3$. In this work, we first show that the $L^\infty$-boundedness of the system can be reduced to the boundedness of $\|u(\cdot,t)\|_{L^{\frac{n}{4}+ε}(Ω)}$ for some $ε>0$ alone, and then, for $n\geq 4$, if the initial data $\|u_0\|_{L^{\frac{n}{4}}}$, $\|z_0\|_{L^\frac{n}{2}}$ and $\|\nabla v_0 \|_{L^n}$ are sufficiently small, we are able to establish the $L^\infty$-boundedness of the system. Furthermore, we show that boundedness implies exponential convergence with explicit convergence rate, which resolves the open problem left in \cite{FIWY16}.

math.AP↗

arXiv2015

A class of chemotaxis systems with growth source and nonlinear secretion

In this paper, we are concerned with a class of parabolic-elliptic chemotaxis systems encompassing the prototype $$\left\{\begin{array}{lll} &u_t = \nabla\cdot(\nabla u-χu\nabla v)+f(u), & x\in Ω, t>0, \\[0.2cm] &0= Δv -v+u^κ, & x\in Ω, t>0 \end{array}\right. $$ with nonnegative initial condition for $u$ and homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset \mathbb{R}^n(n\geq 2)$, where $χ>0$, $κ>0$ and $f$ is a smooth growth source satisfying $f(0)\geq 0$ and $$ f(s)\leq a-bs^θ, \quad s\geq 0, \text{with some} a\geq 0, b>0, θ>1. $$ Firstly, it is shown, either $$ κ<\frac{2}{n}\quad \& \quad f\equiv 0, $$ or $$θ>κ+1, $$ or $$ θ-κ=1, \ \ b\geq \frac{(κn-2)}{κn}χ, \eqno(*) $$ that the corresponding initial-value problem admits a unique classical solution that is uniformly bounded in space and time. Our proof is elementary and semigroup-free. Whilst, with the particular choices $θ=2$ and $κ=1$, Tello and Winkler \cite{TW07} use sophisticated estimates via the Neumann heat semigroup to obtain the global boundedness under the strict inequality in ($\ast$). Thereby, we improve their results to the "borderline" case $b=(κn-2)/(κn)χ$ in this regard. Next, for an unbounded range of $χ$, the system is shown to exhibit pattern formations, and, the emerging patterns are shown to converge weakly in $ L^θ(Ω)$ to some constants as $χ\rightarrow \infty$. While, for small $χ$ or large damping $b$, precisely $b>2χ$ if $f(u)=u(a-bu^κ)$ for some $a, b>0$, we show that the system does not admit pattern formation and the large time behavior of solutions is comparable to its associated ODE+algebraic system.

math.AP↗