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.

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 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.

Global Existence and Blow-Up in Finite Time of Solutions to One Kind of Reaction-Diffusion Educations with Neumann Boundary Conditions

2014 ◽  
Vol 971-973 ◽  
pp. 1017-1020
Author(s):  
Jun Zhou Shao ◽  
Ji Jun Xu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Blowing Up ◽  
Processing Techniques

This paper deals with the properties of one kind of reaction-diffusion equations with Neumann boundary conditions based on the comparison principles. The relations of parameter and the situation of the coupled about equations are used to construct the global existent super-solutions and the blowing-up sub-solutions, and then we obtain the conditions of the global existence and blow-up in finite time solutions with the processing techniques of inequality.

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.

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