Signal-dependent Sensitivity Research Articles

Boundedness and asymptotic stability of solutions in an alopecia areata chemotaxis system with signal-dependent sensitivity

Boundedness and asymptotic stability of solutions in an alopecia areata chemotaxis system with signal-dependent sensitivity

Existence of global weak solution to tumor chemotaxis competition systems with loop and signal dependent sensitivity

This article examines the weak solution of a fully parabolic chemotaxis-competition system with loop and signal-dependent sensitivity. The system is subject to homogeneous Neumann boundary conditions within an open, bounded domain \(\Omega\subset\mathbb{R}^n\), where \(n\geq 1\) and \(\partial\Omega\) is smooth. We assume that the parameters in the system are positive constants. Additionally, the initial data \((u_{10}, u_{20}, v_{10}, v_{20})\in L^2(\Omega)\times L^2(\Omega) \times W^{1,2}(\Omega)\times W^{1,2}(\Omega)\) are non-negative. The existence of a weak solution to the problem is established using energy inequality method. For more information see https://ejde.math.txstate.edu/Volumes/2024/56/abstr.html

Long time asymptotics of small mass solutions for a chemotaxis-consumption system involving prescribed signal concentrations on the boundary

This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity χ=χ(c) under no-flux/Dirichlet boundary conditions. For general χ which may allow singularities at c=0, the global existence and boundedness of radial large data solutions are established in dimensions d≥2. In particular, when χ(c)=1, we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with χ(c)=c, a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial (d=2) and radial (d≥2) solutions at suitably small mass levels.

Global existence in a two-species chemotaxis system with signal-dependent sensitivity and logistic source

Global existence in a two-species chemotaxis system with signal-dependent sensitivity and logistic source

Boundedness criteria for a class of indirect (and direct) chemotaxis-consumption models with signal-dependent sensitivity in high dimensions

This paper is concerned with an indirect chemotaxis-consumption model with sensitivity function χ(v). Frassu and Viglialoro (Appl. Math. Lett.; 2022; 132; 108108) showed global existence and boundedness in the case that χ is a constant. However, there is a room for improvement of the main claim. This paper improves the result and extends it in the case that χ depends on v.

Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system ut=∇⋅(D(u)∇u)−∇⋅(G(u)χ(v)∇v)+∇⋅(H(u)ξ(w)∇w),x∈Ω,t>0,vt=d1Δv+αu−βv,x∈Ω,t>0,wt=d2Δw+γu−δw,x∈Ω,t>0,under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂Rn(n≥1) is a bounded domain with smooth boundary, d1,d2,α,β,γ,δ>0 are constants. Also, D,G,H∈C2([0,∞)) fulfill that a0(s+1)m−1≤D(s)≤a1(s+1)m−1 with a0,a1>0 and m∈R; G(0)=0, 0≤G(s)≤b0(s+1)q−1 with b0>0 and q<min{2,m+1}; H(0)=0, 0≤H(s)≤c0(s+1)r−1 with c0>0 and r<min{2,m+1}, and χ,ξ satisfy that 0≤χ(s)≤χ0sk1 with χ0>0 and k1>1; 0≤ξ(s)≤ξ0sk2 with ξ0>0 and k2>1. Global existence and boundedness in the case that w=0 were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.

Fully parabolic chemotaxis-competition system with loop and signal dependent sensitivity

Fully parabolic chemotaxis-competition system with loop and signal dependent sensitivity

Equilibration In A Two-Species-Two-Chemicals Chemotaxis-Competition System

This paper deals with the two-species--two-chemicals chemotaxis-competition system with signal-dependent sensitivity, where Ω is a bounded domain in Rn (n ≥ 2) with smooth boundary, χ1,χ2 and μ1,μ2 are constants satisfying some conditions. About this system Tu–Mu–Zheng–Lin (Discrete Contin. Dyn. Syst.;2018;38;3617– 3636) showed global existence and stabilization of solutions under some smallness conditions for χ1 and χ2. Here energy arguments for seeing stabilization in the previous work were based on ideas in Bai–Winkler (Indiana Univ. Math. J.;2016;65;553–583); however, these ideas were recently improved by the first author (Discrete Contin. Dyn. Syst. Ser. S;2020;13;269–278), which implies that the result about stabilization in the previous work seems not to be the best. This paper gives an improve- ment of conditions for stabilization in the previous work. The feature of the proof is to use the Sylvester criterion in deriving energy estimates.

Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

&lt;p style='text-indent:20px;'&gt;In this work, the fully parabolic chemotaxis-competition system with loop&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} &amp;amp;\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\&amp;amp; \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &amp;amp;\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\&amp;amp; \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &amp;amp;\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &amp;amp;\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;is considered under the homogeneous Neumann boundary condition, where &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ x\in\Omega, t&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \Omega\subset \mathbb{R}^{n} (n\leq 3) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mu_1, \mu_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are sufficiently large, then the system possesses a unique and global classical solution for &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ n\leq 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Specifically, when &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ n = 2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the global boundedness can be attained without any constraints on &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \mu_1, \mu_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity

&lt;p style='text-indent:20px;'&gt;In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diffusion and sensitivity as follows&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE8011"&gt; \begin{document}$ \begin{cases} u_t = \nabla\cdot(\gamma(v)\nabla u-\chi(v)u\nabla v)+\alpha u F(w) +\theta u-\beta u^2, &amp;amp;x\in \Omega, \; \; t&amp;gt;0,\\ v_t = D\Delta v+u-v,&amp;amp; x\in \Omega, \; \; t&amp;gt;0,\\ w_t = \Delta w-uF(w),&amp;amp; x\in \Omega, \; \; t&amp;gt;0,\\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0,&amp;amp;x\in \partial\Omega, \; \; t&amp;gt;0,\\ u(x,0) = u_0(x), v(x,0) = v_0(x),w(x,0) = w_0(x), &amp;amp; x\in\Omega, \end{cases} \;\;(*)$ \end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;in a bounded domain &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with smooth boundary, where &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \alpha,\beta, D $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are positive constants, &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \theta\in \mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \nu $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; denotes the outward normal vector of &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \partial \Omega $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The functions &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \chi(v),\gamma(v) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ F(v) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; satisfy&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;● &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ (\gamma(v),\chi(v))\in [C^2[0,\infty)]^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \gamma(v)&amp;gt;0,\gamma'(v)&amp;lt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \frac{|\chi(v)|+|\gamma'(v)|}{\gamma(v)} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is bounded;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;● &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ F(w)\in C^1([0,\infty)), F(0) = 0,F(w)&amp;gt;0 \ \mathrm{in}\; (0,\infty)\; \mathrm{and}\; F'(w)&amp;gt;0 \ \mathrm{on}\ \ [0,\infty). $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;We first prove that the existence of globally bounded solution of system (*) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ (u,v,w) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; will converge to &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ (0,0,w_*) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ L^\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with some &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$ w_*\geq0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; as time tends to infinity in the case of &lt;inline-formula&gt;&lt;tex-math id="M16"&gt;\begin{document}$ \theta\leq 0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, while if &lt;inline-formula&gt;&lt;tex-math id="M17"&gt;\begin{document}$ \theta&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the solution &lt;inline-formula&gt;&lt;tex-math id="M18"&gt;\begin{document}$ (u,v,w) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; will asymptotically converge to &lt;inline-formula&gt;&lt;tex-math id="M19"&gt;\begin{document}$ (\frac{\theta}{\beta},\frac{\theta}{\beta},0) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in &lt;inline-formula&gt;&lt;tex-math id="M20"&gt;\begin{document}$ L^\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-norm provided &lt;inline-formula&gt;&lt;tex-math id="M21"&gt;\begin{document}$ D&amp;gt;\max\limits_{0\leq v\leq \infty}\frac{\theta|\chi(v)|^2}{16\beta^2\gamma(v)} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

A new result for asymptotic stability in a two-species chemotaxis model with signal-dependent sensitivity

This paper deals with a two-species chemotaxis model with signal-dependent sensitivity ut=Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u),x∈Ω,t>0,vt=Δv−∇⋅(vχ2(w)∇w)+μ2v(1−v),x∈Ω,t>0,wt=dΔw+h(u,v,w),x∈Ω,t>0,under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥1), where the parameters μ1>0, μ2>0 and d>0, and the chemotactic function χi(w) (i=1,2) are smooth and satisfy some conditions. The global existence and asymptotic behavior of solutions to this system have been studied under some conditions (Mizukami and Yokota, 2016; Mizukami, 2016). The purpose of this work is to improve the conditions assumed in (Mizukami and Yokota, 2016; Mizukami, 2016) for asymptotic behavior.

Boundedness and Stability in a Chemotaxis-Growth Model with Indirect Attractant Production and Signal-Dependent Sensitivity

We study the chemotaxis-growth system with signal-dependent sensitivity function and logistic source $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\nabla \cdot \bigl(u\chi (v)\nabla v\bigr)+\mu u(1-u), &x \in \varOmega ,\ t>0, \\ v_{t}=d\Delta v+h(v,w), &x\in \varOmega ,\ t>0, \\ \tau w_{t}=-\delta w+u, &x\in \varOmega ,\ t>0, \\ \end{array}\displaystyle \right . \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)$ , where the parameters $\mu , \tau , \delta >0$ and $d\geq 0$ , the functions $\chi (v)$ , $h(v,w)$ satisfying some conditions represent the chemotactic sensitivity and the balance between the production and degradation of the chemical signal which relies explicitly on the living organisms, respectively. In the case that $\chi (v)\equiv 1$ , $d=1$ and $h(v,w)=-v+w$ , Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) asserted global existence of bounded solutions for arbitrary $\mu >0$ and established asymptotic behavior of solutions to the mentioned system under the condition $\mu >\frac{1}{8\delta ^{2}}$ in the three dimensional space. The purpose of the present paper is to investigate the global existence and boundedness of classical solutions and to improve the condition assumed in Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) by extending the previous method for obtaining asymptotic stability. Consequently, the range of $\mu $ is extended in the present paper.

Global existence and asymptotic behavior of the fractional chemotaxis system with signal-dependent sensitivity

In this paper, we study a fractional chemotaxis model with signal-dependent sensitivity and the source term, which involves two parabolic equations with fractional diffusions. We develop a suitable mathematical framework for a unified treatment of the existence and decay estimates. Based on an appropriate functional space, we show the existence, uniqueness and temporal decay of the global classical solutions to the problem simultaneously under the smallness initial assumptions.

Global in Time and Bounded Solutions to a Parabolic–Elliptic Chemotaxis System with Nonlinear Diffusion and Signal-Dependent Sensitivity

This paper deals with a system of two coupled partial differential equations arising in chemotaxis, involving nonlinear diffusion and nonlinear and signal-dependent sensitivity. Depending on the interplay between such nonlinearities, we establish the existence of global classical solutions which are uniformly bounded in time. Precisely, we study the zero-flux chemotaxis-system $$\varOmega $$ being a bounded and smooth domain of $$\mathbb {R}^n$$ , $$n\ge 1$$ , and where $$m,\alpha \in \mathbb {R}$$ , with $$\alpha \le \max \{m,\frac{m+1}{2}\}$$ . Additionally, $$0<\chi \in C^1((0,\infty ))$$ obeys the inequality $$\chi (s)\le \frac{\chi _0}{s^k}$$ , for some $$\chi _0>0, k\ge 1$$ and all $$s>0$$ . We prove that for any nonnegative and properly regular initial data u(x, 0), the initial-boundary value problem associated to ( $$\Diamond $$ ) admits a unique globally bounded classical solution, provided some smallness assumptions on $$\chi _0$$ are satisfied. In addition, in this article we compare our results with those achieved in the recent paper (Wang et al. in J Differ Equ 263(5):2851–2873, 2017); we will emphasize how the employment of independent techniques used to solve problem ( $$\Diamond $$ ) may lead to complementary conclusions.

Stabilization in the Keller–Segel system with signal-dependent sensitivity

This paper deals with the Keller–Segel system with signal-dependent sensitivity where and S is a given function generalizing the sensitivity , k>1, , and shows exponential convergence of global classical solutions under an additional smallness condition for .

Solvability of a Keller–Segel system with signal-dependent sensitivity and essentially sublinear production

ABSTRACT In this paper, we consider the zero-flux chemotaxis system in a smooth and bounded domain Ω of . The chemotactic sensitivity χ is a general nonnegative function from while g, the production of the chemical signal v, belongs to and satisfies , for all , and It is established that no chemotactic collapse for the cell distribution u occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data emanates a unique pair of global and uniformly bounded functions which classically solve the corresponding initial boundary value problem. Finally, we illustrate the range of dynamics present within the chemotaxis system by means of numerical simulations.

Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

This paper deals with the two-species chemotaxis-competition system \begin{document}$\begin{equation*} \begin{cases} u_t = d_1Δ u - \nabla · (u χ_1(w)\nabla w) +μ_1 u(1-u-a_1 v)&{\rm in} Ω × (0, ∞), v_t = d_2Δ v - \nabla · (v χ_2(w)\nabla w) +μ_2 v(1-a_2u-v)&{\rm in} Ω × (0, ∞), w_t = d_3Δ w + α u + β v - γ w&{\rm in} Ω × (0, ∞), \end{cases} \end{equation*}$ \end{document} where \begin{document}$Ω$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with smooth boundary \begin{document}$\partial Ω$\end{document} , \begin{document}$n≥ 2$\end{document} ; \begin{document}$χ_i$\end{document} are functions satisfying some conditions. About this problem, Bai-Winkler [ 1 ] first obtained asymptotic stability in (1) under some conditions in the case that \begin{document}$a_1, a_2∈ (0, 1)$\end{document} . Recently, the conditions assumed in [ 1 ] were improved ([ 6 ]); however, there is a gap between the conditions assumed in [ 1 ] and [ 6 ]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that \begin{document}$a_1, a_2∈ (0, 1)$\end{document} .

Global well-posedness and asymptotic stabilization for chemotaxis system with signal-dependent sensitivity

A fully parabolic chemotaxis systemut=Δu−∇⋅(uχ(v)∇v),vt=Δv−v+u, in a smooth bounded domain Ω⊂RN, N≥2 with homogeneous Neumann boundary conditions is considered, where the non-negative chemotactic sensitivity function χ satisfies χ(v)≤μ(a+v)−k, for some a≥0 and k≥1. It is shown that a novel type of weight function can be applied to a weighted energy estimate for k>1. Consequently, the range of μ for the global existence and uniform boundedness of classical solutions established by Mizukami and Yokota [23] is enlarged. Moreover, under a convexity assumption on Ω, an asymptotic Lyapunov functional is obtained and used to establish the asymptotic stability of spatially homogeneous equilibrium solutions for k≥1 under a smallness assumption on μ. In particular, when χ(v)=μ/v and N<8, it is shown that the spatially homogeneous steady state is a global attractor whenever μ≤1/2.

Global existence and boundedness in a chemotaxis–haptotaxis system with signal-dependent sensitivity

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity{ut=Δu−∇⋅(χ(v)u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂Rn(n≥3) is a bounded domain with smooth boundary, ξ,μ>0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4], Tao [10], Tao and Winkler [11]); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system.

Global boundedness in a fully parabolic quasilinear chemotaxis system with singular sensitivity

In this paper we study the global boundedness of solutions to the quasilinear fully parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇φ(v)), vt=Δv−v+u, where bounded domain Ω⊂Rn (n≥2) subject to the non-flux boundary conditions, the diffusivity fulfills D(u)=a0(u+1)−α with a0>0 and α≥0, while the density-signal governed sensitivity satisfies 0≤S(u)≤b0(u+1)β and 0<φ′(v)≤χvk for b0,χ>0 and β,k∈R. It is shown that the solution is globally bounded provided α+β<1 and k≤1. This result demonstrates the effect of signal-dependent sensitivity on the blow-up prevention.