arXiv subjects

Zhi-an Wang

Publications and source records attributed to Zhi-an Wang.

3 recordsLinked to original sources

arXiv2024

Convergence of boundary layers of chemotaxis models with physical boundary conditions I: degenerate initial data

The celebrated experiment of Tuval et al. \cite{tuval2005bacterial} showed that the bacteria living a water drop can form a thin layer near the air-water interface, where a so-called chemotaxis-fluid system with physical boundary conditions was proposed to interpret the mechanism underlying the pattern formation alongside numerical simulations. However, the rigorous proof for the existence and convergence of the boundary layer solutions to the proposed model still remains open. This paper shows that the model with physical boundary conditions proposed in \cite{tuval2005bacterial} in one dimension can generate boundary layer solution as the oxygen diffusion rate $\varepsilon>0$ is small. Specifically, we show that the solution of the model with $\varepsilon>0$ will converge to the solution with $\varepsilon=0$ (outer-layer solution) plus the boundary layer profiles (inner-layer solution) with a sharp transition near the boundary as $ \varepsilon \rightarrow 0$. There are two major difficulties in our analysis. First, the global well-posedness of the model is hard to prove since the Dirichlet boundary condition can not contribute to the gradient estimates needed for the cross-diffusion structure in the model. Resorting to the technique of taking anti-derivative, we remove the cross-diffusion structure such that the Dirichlet boundary condition can facilitate the needed estimates. Second, the outer-layer profile of bacterial density is required to be degenerate at the boundary as $ t \rightarrow 0 ^{+}$, which makes the traditional cancellation technique incapable. Here we employ the Hardy inequality and delicate weighted energy estimates to overcome this obstacle and derive the requisite uniform-in-$\varepsilon$ estimates allowing us to pass the limit $\varepsilon \to 0$ to achieve our results.

math.AP↗

arXiv2023

Novel Spatial Profiles of Population Distribution of Two Diffusive SIS Epidemic Models with Mass Action Infection Mechanism and Small Movement Rate for the Infected Individuals

In this paper, we are concerned with two SIS epidemic reaction-diffusion models with mass action infection mechanism of the form $SI$, and study the spatial profile of population distribution as the movement rate of the infected individuals is restricted to be small. For the model with a constant total population number, our results show that the susceptible population always converges to a positive constant which is indeed the minimum of the associated risk function, and the infected population either concentrates at the isolated highest-risk points or aggregates only on the highest-risk intervals once the highest-risk locations contain at least one interval. In sharp contrast, for the model with a varying total population number which is caused by the recruitment of the susceptible individuals and death of the infected individuals, our results reveal that the susceptible population converges to a positive function which is non-constant unless the associated risk function is constant, and the infected population may concentrate only at some isolated highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or occupy the whole habitat, depending on the behavior of the associated risk function and even its smoothness at the highest-risk locations. Numerical simulations are performed to support and complement our theoretical findings.

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↗