Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.

Boundedness and Asymptotic Behavior to a Chemotaxis System with Indirect Signal Generation and Singular Sensitivity

In this paper, we consider the following indirect signal generation and singular sensitivity n t = Δ n + χ ∇ ⋅ n / φ c ∇ c , x ∈ Ω , t > 0 , c t = Δ c − c + w , x ∈ Ω , t > 0 , w t = Δ w − w + n , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R N N = 2 , 3 with smooth boundary ∂ Ω . Under the nonflux boundary conditions for n , c , and w , we first eliminate the singularity of φ c by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.

Asymptotic profile of a two-dimensional Chemotaxis–Navier–Stokes system with singular sensitivity and logistic source

This paper is concerned with a spatially two-dimensional version of a chemotaxis system with logistic cell proliferation and death, for a singular tactic response of standard logarithmic type, and with interaction with a surrounding incompressible fluid through transport and buoyancy. Systems of this form are of significant relevance to the understanding of chemotaxis-fluid interaction, but the rigorous knowledge of their qualitative properties is yet far from complete. In this direction, using the conditional energy functional method, the present work provides some interesting contributions by establishing results on global boundedness, and especially on large time stabilization toward homogeneous equilibria, under mild assumptions on the initial data and appropriate conditions on the strength of the damping death effects.

A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity

<p style='text-indent:20px;'>This paper deals with the dynamical properties of the quasilinear parabolic-parabolic chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(\frac{u}{v} \nabla v)+\mu u- \mu u^{2}, \, \, \, &amp;x\in\Omega, \, \, \, t&gt;0, \\ v_{t} = \Delta v-v+u, &amp;x\in\Omega, \, \, \, t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a convex bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\geq2 $\end{document}</tex-math></inline-formula>, with smooth boundary. <inline-formula><tex-math id="M3">\begin{document}$ \chi&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ D(u) $\end{document}</tex-math></inline-formula> is supposed to satisfy the behind properties</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha&gt;0. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown that there is a positive constant <inline-formula><tex-math id="M6">\begin{document}$ m_{*} $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for all <inline-formula><tex-math id="M7">\begin{document}$ t\geq0 $\end{document}</tex-math></inline-formula>. Moreover, we prove that the solution is globally bounded. Finally, it is asserted that the solution exponentially converges to the constant stationary solution <inline-formula><tex-math id="M8">\begin{document}$ (1, 1) $\end{document}</tex-math></inline-formula>.</p>