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.

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 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 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 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.

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.

scholarly journals Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Rong Zhang ◽  
Liangchen Wang
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Large Time ◽  
Neumann Boundary ◽  
Rates Of Convergence ◽  
Large Time Behavior ◽  
Global Dynamics ◽  
Time Behavior ◽  
Chemical Signalling ◽  
Chemotaxis System

<p style='text-indent:20px;'>This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\, x\in \Omega,\, t&gt;0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t&gt;0,\\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)-\chi_3\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), \,x\in \Omega,\,t&gt;0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula>, where the parameters <inline-formula><tex-math id="M3">\begin{document}$ a_1,a_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \chi_1, \chi_2, \chi_3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula> are positive constants. We first showed some conditions between <inline-formula><tex-math id="M6">\begin{document}$ \frac{\chi_1}{\mu_1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \frac{\chi_2}{\mu_2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \frac{\chi_3}{\mu_2} $\end{document}</tex-math></inline-formula> and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.</p>

scholarly journals Global bounded solution of the higher-dimensional forager–exploiter model with/without growth sources

2020 ◽  
Vol 30 (07) ◽  
pp. 1297-1323 ◽  
Author(s):  
Jianping Wang ◽  
Mingxin Wang
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Classical Solution ◽  
Neumann Boundary Condition ◽  
Bounded Solution ◽  
Neumann Boundary ◽  
Homogeneous Neumann Boundary Condition ◽  
Higher Dimensional ◽  
Degradation Rates ◽  
Growth Restrictions

This paper concerns with the global existence and boundedness of classical solution of the higher-dimensional forager–exploiter model with homogeneous Neumann boundary condition and nonnegative initial data. For cases where there are no forager and exploiter growth sources, it will be shown that if either the initial data and the production rate of nutrient are small or the taxis effects are small, then the classical solution exists globally and is bounded. For the case that only the forager has growth source, when [Formula: see text], it will be shown that if the taxis effect of exploiter is small then the classical solution exists globally and is bounded. For the case that both the forager and exploiter have growth restrictions, when [Formula: see text], we find a condition for the logistic degradation rates that ensures the global existence and boundedness of the classical solution.

scholarly journals Blow-Up in a Slow Diffusive -Laplace Equation with the Neumann Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chengyuan Qu ◽  
Bo Liang
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Laplace Equation ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Initial Energy ◽  
Neumann Boundary Conditions ◽  
Natural Energy

We study a slow diffusive -Laplace equation in a bounded domain with the Neumann boundary conditions. A natural energy is associated to the equation. It is shown that the solution blows up in finite time with the nonpositive initial energy, based on an energy technique. Furthermore, under some assumptions of initial data, we prove that the solutions with bounded initial energy also blow up.

Sign in / Sign up

Export Citation Format

Share Document