Large time behavior for a viscous Hamilton–Jacobi equation with Neumann boundary condition

This paper is concerned with large time behavior of solutions to the compressible Navier–Stokes equations in an infinite layer of [Formula: see text] under slip boundary condition. It is shown that if the initial data is sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one-dimensional diffusion waves.

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

Electronic Research Archive ◽

10.3934/era.2021086 ◽

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

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop<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>0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t>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>0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t>0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \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.

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Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction

Discrete and Continuous Dynamical Systems - Series B ◽

10.3934/dcdsb.2021255 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Chao Liu ◽

Bin Liu

Keyword(s):

Asymptotic Behavior ◽

Large Time ◽

Neumann Boundary ◽

Large Time Behavior ◽

Time Behavior ◽

Smooth Bounded Domain ◽

Predator Prey ◽

Predator Prey Model ◽

Pursuit Evasion ◽

Predator Model

In this paper, we study the prey-predator model with indirect pursuit-evasion interaction defined on a smooth bounded domain with homogeneous Neumann boundary conditions. We obtain the globa existence and boundedness of the classical solution of the model by estimating <inline-formula><tex-math id="M1">\begin{document}$ L^{p} $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>, and we also show the large time behavior and convergence rate of the solution.

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