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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jie Zhao
Keyword(s):  
Boundary Conditions ◽  
Positive Constant ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Logistic Source ◽  
Chemotaxis Model ◽  
Singular Sensitivity ◽  
Dynamical Properties ◽  
Chemotaxis System ◽  
Quasilinear Parabolic

<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>

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

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Heping Ma
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Global Classical Solution ◽  
Singular Sensitivity ◽  
Chemotaxis System

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.

scholarly journals Global Existence and Blow-up of Classical Solution for an Attraction-repulsion Chemotaxis System with Logistic Source

2019 ◽  
pp. 1-16
Author(s):  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Finite Time ◽  
Blow Up ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Elliptic Type ◽  
Logistic Source ◽  
Chemotaxis System

We consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic-elliptic type with logistic source under homegeneous Neumann boundary conditions in a bounded domain `\Omega\subset R^{n}(n\geq2)` with smooth boundary, where`D(u)\geq c_{D}(u+1)^{m-1}` with `m\geq1`and `c_{D}>0`, `f(u)\leq a-bu^{\eta}` with `\eta>1`.{ We show two cases that the system admits a uniqueglobal bounded classical solution depending on `0\leq S(u)\leq C_{s}(u+1)^{q}, 0\leq F(u)\leq C_{F}(u+1)^{g}` by Gagliardo-Nirenberg inequality.For specific `D(u),S(u),F(u)` with logistic source for `\eta>1` and `n=2`, we establish the finite time blow-up conditions forsolutions that the finite time blow-up occurs at `x_{0}\in\Omega` whenever `\int_{\Omega}u_{0}(x)dx>\frac{8\pi}{\chi\alpha-\xi\gamma}`with `\chi\alpha-\xi\gamma>0`, under `\int_{\Omega}u_{0}(x)|x-x_{0}|^{2}dx` sufficiently small.

scholarly journals Global Existence and Boundedness of a Two-Competing-Species Chemotaxis Model

2019 ◽  
pp. 1-5
Author(s):  
Liangying Miao
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Comparison Principle ◽  
Bounded Solution ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Competing Species ◽  
Chemotaxis Model

In this paper, we consider the following fully parabolic two-competing-species chemotaxis model $$\left\{\begin{array}{ll}\displaystyle u_{1t}=\Delta{u_{1}}-\chi \nabla\cdot(u_{1}\nabla{v_{1}})+\mu_{1}u_{1}(1-u_{1}-e_{1}u_{2}),&x\in\Omega,~ t>0,\\\displaystyle u_{2t}=\Delta{u_{2}}-\xi\nabla\cdot(u_{2}\nabla{v_{2}})+\mu_{2}u_{2}(1-e_{2}u_{1}-u_{2}),&x\in\Omega,~t>0,\\\displaystyle v_{1t}=\Delta{v_{1}}+u_{1}- v_{1},&x\in\Omega,~ t>0, \\\displaystyle v_{2t}=\Delta{v_{2}}+u_{2}- v_{2},&x\in\Omega,~ t>0\end{array}\right.$$ under homogeneous Neumann boundary conditions, where Ω ⊂ ℝn  (n≥3) is a convex bounded domain with smooth boundary. Relying on a comparison principle, we show that the problem possesses a uniqueglobal bounded solution if μ1 and μ2 are large enough.

An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization

2020 ◽  
pp. 1-21 ◽  
Author(s):  
Wenbin Lv ◽  
Qingyuan Wang
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Neumann Boundary ◽  
Logistic Source ◽  
Global Classical Solution ◽  
Higher Dimensional ◽  
Image Position ◽  
Chemotaxis System

Abstract This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as $$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta v-v + u,\quad x\in \Omega ,\;t > 0.} $$ It is shown that whenever ρ ∈ ℝ, μ > 0 and $$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$ then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium $$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$ as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Jie Wu ◽  
Li Zhao ◽  
Heping Pan
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Rates Of Convergence ◽  
Signal Generation ◽  
Heat Semigroup ◽  
Singular Sensitivity ◽  
Chemotaxis System ◽  
Global Boundedness

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.

EXISTENCE AND BEHAVIOR OF SOLUTION TO SOME CHEMOTAXIS MODEL WITH INTERACTING SPECIES

2010 ◽  
Vol 03 (03) ◽  
pp. 367-382 ◽  
Author(s):  
JUN-FENG LI ◽  
WEI-AN LIU
Keyword(s):  
Boundary Conditions ◽  
Solution Method ◽  
Existence Of Solution ◽  
Chemotaxis Model ◽  
Interacting Species ◽  
Initial And Boundary Conditions ◽  
Chemotaxis System ◽  
And Behavior

In this paper, we study a chemotaxis system involving two interacting species as follows: [Formula: see text] with some initial and boundary conditions. We prove the existence of solution to the problem whose behavior is concerned with reference to some systems through sup-sub-solution method.

scholarly journals Global well-posedness in a chemotaxis system with oxygen consumption

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xujie Yang
Keyword(s):  
Smooth Boundary ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Liquid Environment ◽  
Bounded Convex Domain ◽  
Initial Boundary Value ◽  
Chemotaxis System ◽  
Bounded Convex ◽  
Uniformly Bounded

<p style='text-indent:20px;'>Motivated by the studies of the hydrodynamics of the tethered bacteria <i>Thiovulum majus</i> in a liquid environment, we consider the following chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{split} &amp; n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &amp;x\in \Omega, t&gt;0, \ &amp; c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &amp;x\in \Omega, t&gt;0\ \end{split} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded convex domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $\end{document}</tex-math></inline-formula> with smooth boundary. For any given fluid <inline-formula><tex-math id="M2">\begin{document}$ {\bf u} $\end{document}</tex-math></inline-formula>, it is proved that if <inline-formula><tex-math id="M3">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if <inline-formula><tex-math id="M4">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>, such solution still exists under the additional condition that <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $\end{document}</tex-math></inline-formula>.</p>

Sign in / Sign up

Export Citation Format

Share Document