scholarly journals ASYMPTOTIC DYNAMICS OF A NEW MECHANOCHEMICAL MODEL IN BIOLOGICAL PATTERNS

2017 ◽  
Vol 22 (2) ◽  
pp. 252-269 ◽  
Author(s):  
Aibo Liu ◽  
Changchun Liu
Keyword(s):  
Boundary Conditions ◽  
Existence Theorem ◽  
Bounded Domain ◽  
Space Dimension ◽  
Neumann Boundary ◽  
Global Attractors ◽  
Regularity Estimates ◽  
Asymptotic Dynamics ◽  
Mechanochemical Model ◽  
Biological Patterns

In this paper, we prove the existence of attractor for a new mechanochemical model with Neumann boundary conditions on a bounded domain of space dimension n ≤ 3. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the mechanochemical model possesses a global attractor and Hk attractor.

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 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 Dirichlet and Neumann boundary conditions for the p-Laplace operator: what is in between?

2012 ◽  
Vol 142 (5) ◽  
pp. 975-1002 ◽  
Author(s):  
Ralph Chill ◽  
Mahamadi Warma
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Lower Semicontinuous ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Lipschitz Continuous ◽  
Type Boundary ◽  
Continuous Boundary

Let p ∈ (1, ∞) and let Ω ⊆ ℝN be a bounded domain with Lipschitz continuous boundary. We characterize on L2(Ω) all order-preserving semigroups that are generated by convex, lower semicontinuous, local functionals and are sandwiched between the semigroups generated by the p-Laplace operator with Dirichlet and Neumann boundary conditions. We show that every such semigroup is generated by the p-Laplace operator with Robin-type boundary conditions.

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 dynamics of solutions for a sixth-order parabolic equation describing continuum evolution of film-free surface

2022 ◽  
Vol 27 (1) ◽  
pp. 19-37
Author(s):  
Ning Duan ◽  
Xiaopeng Zhao
Keyword(s):  
Free Surface ◽  
Parabolic Equation ◽  
Diffusion Equation ◽  
Existence Theorem ◽  
Global Attractor ◽  
Global Attractors ◽  
Sixth Order ◽  
Global Dynamics ◽  
Regularity Estimates ◽  
Iteration Technique

This paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces H3(Ω) and fractional-order spaces Hk(Ω), where 0 ≤ k < ∞.

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.

scholarly journals Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 65 ◽  
Author(s):  
Jean-Daniel Djida ◽  
Arran Fernandez
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Degenerate Elliptic Equation ◽  
Extension Problem ◽  
Regularity Estimates ◽  
Degenerate Elliptic

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.

Study of fluid flow through a channel deformed by piezoelement

2018 ◽  
Vol 13 (3) ◽  
pp. 1-10 ◽  
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh Nasibullaeva ◽  
O.V. Darintsev
Keyword(s):  
Fluid Flow ◽  
Pressure Drop ◽  
Boundary Conditions ◽  
Flow Rate ◽  
Contact Surface ◽  
Piezoelectric Element ◽  
Neumann Boundary ◽  
Differential Pressure ◽  
Physical Parameters ◽  
Flow Through

The flow of a liquid through a tube deformed by a piezoelectric cell under a harmonic law is studied in this paper. Linear deformations are compared for the Dirichlet and Neumann boundary conditions on the contact surface of the tube and piezoelectric element. The flow of fluid through a deformed channel for two flow regimes is investigated: in a tube with one closed end due to deformation of the tube; for a tube with two open ends due to deformation of the tube and the differential pressure applied to the channel. The flow rate of the liquid is calculated as a function of the frequency of the deformations, the pressure drop and the physical parameters of the liquid.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions

2020 ◽  
Vol 28 (2) ◽  
pp. 237-241
Author(s):  
Biljana M. Vojvodic ◽  
Vladimir M. Vladicic
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions

AbstractThis paper deals with non-self-adjoint differential operators with two constant delays generated by {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})}, where {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials {q_{j}} are real-valued functions, {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions {y(0)=y(\pi)=0} and the remaining two under boundary conditions {y(0)=y^{\prime}(\pi)=0}.

Sign in / Sign up

Export Citation Format

Share Document