Singular Sensitivity Research Articles

Existence and stability of boundary spike layer solutions of an attractive chemotaxis model with singular sensitivity and nonlinear consumption rate of chemical stimuli

Existence and stability of boundary spike layer solutions of an attractive chemotaxis model with singular sensitivity and nonlinear consumption rate of chemical stimuli

Global boundedness in a parabolic-elliptic chemotaxis system with singular sensitivity and sublinear production

Global boundedness in a parabolic-elliptic chemotaxis system with singular sensitivity and sublinear production

Boundedness in a chemotaxis system with weakly singular sensitivity in dimension two with arbitrary sub-quadratic degradation sources

We study the global existence and boundedness of solutions to a chemotaxis system with weakly singular sensitivity and sub-logistic sources in a two-dimensional domain. It has been shown in X. Zhao (2023) [22] that the logistic degradation, −μu2, can prevent blow-up under the largeness assumption on μ. In this paper, we improve the result by replacing the quadratic degradation by sub-logistic one, −μu2lnβ⁡(u+e) with β∈(0,1), and removing the largeness assumption on μ.

Global existence and boundedness in a chemotaxis model with singular sensitivity and nonlocal term

Global existence and boundedness in a chemotaxis model with singular sensitivity and nonlocal term

Boundedness in the higher-dimensional chemotaxis system for Alopecia Areata with singular sensitivity

This paper deals with a fully parabolic chemotaxis system ut=Δu−χ1∇⋅(uw∇w)+w−μ1u2,x∈Ω,t>0,vt=Δv−χ2∇⋅(vw∇w)+w+ruv−μ2v2,x∈Ω,t>0,wt=Δw+u+v−w,x∈Ω,t>0in a bounded domain Ω⊂Rn(n≥2) with smooth boundary, where r>0,χi>0,μi>0(i=1,2). Under the condition μ1≥2r,μ2≥2r,max{χ1,χ2}<2n, it is shown that the corresponding system possesses a globally bounded classical solution, which extends the previous boundedness result from n=2 to n≥2.

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

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].

Global boundedness and asymptotic behavior of the chemotaxis system for alopecia areata with singular sensitivity

This paper is concerned with a three-component chemotaxis system for alopecia areata with singular sensitivity ut=Δu−χ1∇⋅uw∇w+w−μ1u2,x∈Ω,t>0,vt=Δv−χ2∇⋅vw∇w+w+ruv−μ2v2,x∈Ω,t>0,wt=Δw+u+v−w,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ωunder the homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂R2, where the parameters χi, μi(i=1,2) and r are positive. It is showed that if χ1,χ2<52, this system admits a globally bounded classical solution. Furthermore, under the particular conditions of μ1<μ2<3μ1 and r=μ2−μ1, the global bounded solution converges to the steady state (2μ1,2μ1,4μ1) as t→∞.

Global existence, boundedness and asymptotic behavior of classical solutions to a fully parabolic two-species chemotaxis-competition model with singular sensitivity

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics(0.1){ut=Δu−χ1∇⋅(uw∇w)+μ1u(1−u−a1v),t>0,x∈Ω,vt=Δv−χ2∇⋅(vw∇w)+μ2v(1−v−a2u),t>0,x∈Ω,wt=Δw−w+u+v,t>0,x∈Ω,∂u∂ν=∂v∂ν=∂w∂ν=0,t>0,x∈∂Ω,u(0,x)=u0(x),v(0,x)=v0(x),w(0,x)=w0(x),x∈Ω, where Ω⊂RN(N≥2) is a bounded smooth convex domain, and the parameters χ1,χ2,μ1,μ2,a1 and a2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ1,χ2∈(0,12)forN=2,3 or χ1,χ2∈(0,N−2N−1)forN≥4. Moreover, if min⁡{μ1,μ2}>max⁡{χ1,χ2}24, we obtain the uniformly lower bound for w. Finally, when χ1,χ2 are suitably small, it is proved that if 0<a1,a2<1, then the solution (u,v,w) converges to (1−a11−a1a2,1−a21−a1a2,2−a1−a21−a1a2) in L∞ norm as t→∞; if 0<a2<1≤a1, then the solution (u,v,w) converges to (0,1,1) in L∞ norm as t→∞.

Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity

Abstract In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity: u t = ∇ ⋅ ( u θ − 1 ∇ u ) − χ ∇ ⋅ u v ∇ v , x ∈ Ω , t &gt; 0 , v t = Δ v − v + u + g ( x , t ) , x ∈ Ω , t &gt; 0 , ( ∗ ) \left\{\begin{array}{ll}{u}_{t}=\nabla \cdot \left({u}^{\theta -1}\nabla u)-\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right),&amp; x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta v-v+u+g\left(x,t),&amp; x\in \Omega ,\hspace{0.33em}t\gt 0,\\ \end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\ast ) in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model ( ∗ \ast ) if θ &gt; 3 2 \theta \gt \frac{3}{2} and (1.10)–(1.11). This result improves our recent work.

A chemotaxis system with singular sensitivity for burglaries in the higher-dimensional settings: generalized solvability and long-time behavior

A chemotaxis system with singular sensitivity for burglaries in the higher-dimensional settings: generalized solvability and long-time behavior

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

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

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

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

Dermatophytosis in cats: A comprehensive study on diagnostic methods and their accuracy.

Dermatophytosis is a contagious fungal infection that affects mainly cats. It poses significant challenges in veterinary medicine due to its zoonotic potential and impact on animal and public health. Rapid and reliable diagnosis is crucial for preventing the spread of the disease, guiding treatment decisions, and monitoring disease control efforts. Although there are several studies on diagnostic methods in feline dermatophytosis, the comparison between them from the same sample lacks data. The absence of a universally accepted gold standard diagnostic method highlights the need for a multifaceted approach to diagnosing feline dermatophytosis. This study aims to assess the accuracy and efficacy of different diagnostic techniques comprehensively. For this, 48 samples of cats were analyzed by dermoscopy, direct hair examination, fungal culture using various media (Mycosel, Sabouraud, and Dermatophyte Test Medium), and polymerase chain reaction (PCR). Direct examination and dermoscopy yielded unsatisfactory results. Mycosel and Sabouraud were suboptimal. DTM demonstrated superior selectivity, making it the most reliable among traditional methods. PCR was the top performer, exhibiting singular sensitivity, specificity, and accuracy. The study suggests that PCR may be the preferred choice for diagnosing feline dermatophytosis in clinical practice, especially when rapid and accurate results are essential.

Global classical solvability and asymptotic behaviors of a parabolic-elliptic Chemotaxis-type system modeling crime activities

This paper is devoted to a two-dimensional parabolic-elliptic system in crime activities. The main difference between this system and the chemotaxis system with logarithmic singular sensitivity and quadratic logistic source is that the source term of this system is a mixtype quadratic term. The local existence of solutions to its intial boundary valve problem associated with homogeneous Neumann boundary conditions was considered by Rodríguez. The present study indicates that for any properly regular initial data there admits global classical solutions. Moreover, the qualitative behaviors of solutions in large time scales are considered as well.

Can Dirac-type singularities in Keller-Segel systems be ruled out by power-type singular sensitivities?

This paper deals with the chemotaxis system with power-type singular sensitivities: ut=Δu−χ∇⋅(uvα∇v) and vt=Δv−v+u in a bounded domain Ω⊂R2, with χ>0 and α>0. It is notable that, whenever α=0, this system reduces to the minimal Keller-Segel model; the persistent Dirac-type singularities are known to occur at least in radial parabolic-elliptic setting. In this work, our first result shows that for any χ>0, if eitherα≥14, orα∈(0,14) and ‖u0‖L1 is small appropriately, then the corresponding initial-boundary value problem possesses a global generalized solution (u,v) with the property that u belongs to L1(Ω×(0,T)) for any T>0, which further implies that the general sensitivities 1vα, with α≥14, can rule out the emergence of Dirac-type singularities. Our second result indicates that, if α∈(0,12) and ‖u0‖L1 is small properly, then such global generalized solution becomes bounded and smooth at least eventually, and approaches the spatial equilibria in the large time limit.

Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

Global solutions in a chemotaxis consumption model with singular sensitivity

Global solutions in a chemotaxis consumption model with singular sensitivity

Global Boundedness in a Logarithmic Keller–Segel System

In this paper, we propose a user-friendly integral inequality to study the coupled parabolic chemotaxis system with singular sensitivity under the Neumann boundary condition. Under a low diffusion rate, the classical solution of this system is uniformly bounded. Our proof replies on the construction of the energy functional containing ∫Ω|v|4v2 with v&gt;0. It is noteworthy that the inequality used in the paper may be applied to study other chemotaxis systems.

Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two

This paper deals with the parabolic-elliptic Keller–Segel system with weakly singular sensitivity and logistic source under the homogeneous Neumann boundary: , in a smooth bounded domain , with , . It is proved that the system possesses a globally bounded classical solution for with µ > 0 suitably large, without establishing the uniformly positive bound for v from below. In addition, we give the explicit expression of the upper bound for solution u with respect to the parameters via a recursive argument on α. This concludes that weakly singular sensitivity benefits to obtain the global boundedness of classical solution in dimension two.

Global weak solution for a chemotaxis Navier–Stokes system with [formula omitted]-Laplacian diffusion and singular sensitivity

In this paper, we consider a chemotaxis-Navier–Stokes system with p-Laplacian diffusion and singular sensitivity in a bounded convex domain Ω⊂R3 with smooth boundary. It is shown that under an appropriate hypothesis for p and δ, there exists a locally bounded global weak solution of the corresponding initial–boundary problem.