arXiv subjects

Halil Ibrahim Kurt

Publications and source records attributed to Halil Ibrahim Kurt.

5 recordsLinked to original sources

arXiv2024

Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics

The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i,a_i, b_i, c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ are positive constants. In [25], we proved that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution provided that $\min\{a_1,a_2\}$ is large relative to $χ_1,χ_2$, and $u_0+v_0$ is not small in the case that $(χ_1-χ_2)^2\le \max\{4χ_1,4χ_2\}$ and $u_0+v_0$ is neither small nor big in the case that $(χ_1-χ_2)^2>\max\{4χ_1,4χ_2\}$. In this paper, we proved that (0.1) has a unique positive constant solution $(u^*,v^*,w^*)$, where $$ u^*=\frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2},\quad v^*=\frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, \quad w^*=\fracνμu^*+\fracλμ v^*. $$ We obtain some explicit conditions on $χ_1,χ_2$ which ensure that the positive constant solution $(u^*,v^*,w^*)$ is globally stable in the sense that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0\not \equiv 0$ and $v_0\not \equiv 0$, $$ \lim_{t\to\infty}\Big(\|u(t,\cdot;u_0,v_0)-u^*\|_\infty +\|v(t,\cdot;u_0,v_0)-v^*\|_\infty+\|w(t,\cdot;u_0,v_0)-w^*\|_\infty\Big)=0. $$

math.AP↗

arXiv2024

Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence

This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i$, $a_i$, $b_i$, $ c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ are positive constants. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution $(u(t,x;u_0,v_0),v(t,x;u_0,v_0),w(t,x;u_0,v_0))$ with $u(0,x;u_0,v_0)=u_0(x)$ and $v(0,x;u_0,v_0)=v_0(x)$ provided that $\min\{a_1,a_2\}$ is large relative to $χ_1,χ_2$ and $u_0+v_0$ is not small. Moreover, under the same condition, we prove that \begin{equation*} \limsup_{t\to\infty} \|u(t,\cdot;u_0,v_0)+v(t,\cdot;u_0,v_0)\|_\infty\le M^*, \end{equation*} and \begin{equation*} \liminf_{t\to\infty} \inf_{x\inΩ}(u(t,x,u_0,v_0)+v(t,x;u_0,v_0))\ge m^*, \end{equation*} for some positive constants $M^*,m^*$ independent of $u_0,v_0$, the latter is referred to as combined pointwise persistence.

math.AP↗

arXiv2024

Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

In the current paper, we study stability, bifurcation, and spikes of positive stationary solutions of the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begin{cases} u_t=u_{xx}-χ(\frac{u}{v} v_x)_x+u(a-b u), & 0 0,\cr 0=v_{xx}- μv+ νu, & 0 0 \cr u_x(t,0)=u_x(t,L)=v_x(t,0)=v_x(t,L)=0, & t>0, \tag{1} \end{cases} where $χ$, $a$, $b$, $μ$, $ν$ are positive constants. Among others, we prove there are $χ^*>0$ and $\{χ_k^*\}\subset [χ^*,\infty)$ ($χ^*\in\{χ_k^*\}$) such that the constant solution $(\frac{a}{b},\fracνμ\frac{a}{b})$ of (1) is locally stable when $0<χ<χ^*$ and is unstable when $χ>χ^*$, and under some generic condition, for each $k\ge 1$, a (local) branch of non-constant stationary solutions of (1) bifurcates from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ_k^*$, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions $\{(u(\cdot;χ_n),v(\cdot;χ_n))\}$ of (1) with $χ=χ_n(\to \infty)$ develops spikes at any $x^*$ satisfying $\liminf_{n\to\infty} u(x^*;χ_n)>\frac{a}{b}$. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ^*$ can be extended to $χ=\infty$ and the stationary solutions on this global bifurcation extension are locally stable when $χ\gg 1$ and develop spikes as $χ\to\infty$.

math.AP↗

arXiv2024

Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions

This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(t,x)-b(t,x) u), & x\in Ω,\cr 0=Δv- μv+ νu, & x\in Ω, \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $a(t,x)$ and $b(t,x)$ are positive smooth functions, and $χ$, $μ$ and $ν$ are positive constants. In the very recent paper [25], we proved that for given nonnegative initial function $0\not\equiv u_0\in C^0(\bar Ω)$ and $s\in\mathbb{R}$, (0.1) has a unique globally defined classical solution $(u(t,x;s,u_0),v(t,x;s,u_0))$ with $u(s,x;s,u_0)=u_0(x)$, provided that $a_{\inf}=\inf_{t\in\mathbb{R},x\inΩ}a(t,x)$ is large relative to $χ$ and $u_0$ is not small. In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption that $a_{\inf}$ is large relative to $χ$ and $u_0$ is not small. Among others, we provide some concrete estimates for $\int_Ωu^{-p}$ and $\int_Ωu^q$ for some $p>0$ and $q>\max\{2,N\}$ and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a ``rectangular'' type bounded invariant set (in $L^q$) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution $(u^*(t,x),v^*(t,x))$, which is periodic in $t$ if $a(t,x)$ and $b(t,x)$ are periodic in $t$ and is independent of $t$ if $a(t,x)$ and $b(t,x)$ are independent of $t$.

math.AP↗

arXiv2024

Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current paper is to study the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u^{1+σ}),\quad &x\in Ω\cr 0=Δv-μv+νu,\quad &x\in Ω\quad \cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partialΩ$, $χ$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, $μ,ν$ are positive constants, and $σ\ge 0$. When $σ>0$, we prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar Ω)$, (0.1) has a unique globally defined classical solution $(u_σ(x,t;u_0),v_σ(x,t;u_0))$ with $u_σ(x,0;u_0)=u_0(x)$, which shows that, in any space dimensional setting, strong logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $χ$. In addition, the solutions are shown to be uniformly bounded under the conditions \begin{equation*} a_{\inf}> \begin{cases} \frac{μχ^2}{4}, &\text{if $0< χ\leq 2,$}\\ μ(χ-1), &\text{if $χ>2$.}\\ \end{cases} \end{equation*} When $σ=0$, we show that the classical solution $(u(x,t;u_0,0),v(x,t;u_0,0))$ exists globally and stays bounded provided that both $a(x,t)$ and $u_0(x)$ are not small.

math.AP↗