Asymptotics of solutions to a convection—diffusion equation on the half-line

2000 ◽  
Vol 130 (4) ◽  
pp. 837-853 ◽  
Author(s):  
G. Karch
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Half Line ◽  
Asymptotics Of Solutions ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Homogeneous Neumann Boundary Condition ◽  
Higher Order Terms

We study the behaviour, as t → ∞, of solutions to the convectiondiffusion equation on the half-line with the homogeneous Neumann boundary condition and with bounded initial data. The higher-order terms of the asymptotic expansion in Lp (R+) of solutions are derived.

Positive steady states of the Holling–Tanner prey–predator model with diffusion

2005 ◽  
Vol 135 (1) ◽  
pp. 149-164 ◽  
Author(s):  
Rui Peng ◽  
Mingxin Wang
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Steady States ◽  
Homogeneous Neumann Boundary Condition ◽  
Positive Steady States ◽  
Predator Model

This paper is concerned with the Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We obtain the existence and non-existence of positive non-constant steady states.

On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition

2015 ◽  
Vol 116 ◽  
pp. 19-25 ◽  
Author(s):  
Maria Fărcăşeanu ◽  
Mihai Mihăilescu ◽  
Denisa Stancu-Dumitru
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Homogeneous Neumann Boundary Condition

EFFECT OF CROSS DIFFUSION IN A COMPETITION MODEL WITH STAGE STRUCTURE

2012 ◽  
Vol 05 (06) ◽  
pp. 1250052 ◽  
Author(s):  
LINA ZHANG ◽  
SHENGMAO FU ◽  
PING HU
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Stage Structure ◽  
Competition Model ◽  
Positive Equilibrium ◽  
Cross Diffusion ◽  
Homogeneous Neumann Boundary Condition

The purpose of this paper is to study the effect of cross diffusion in a competition model with stage structure, under homogeneous Neumann boundary condition. It will be shown that cross diffusion cannot only destabilize a uniform positive equilibrium, it can also help diffusion to induce instability of the uniform positive equilibrium. Moreover, stationary patterns can arise from the effect of cross diffusion.

Dynamics Analysis in a Gierer–Meinhardt Reaction–Diffusion Model with Homogeneous Neumann Boundary Condition

2019 ◽  
Vol 29 (09) ◽  
pp. 1930025 ◽  
Author(s):  
Xiang-Ping Yan ◽  
Ya-Jun Ding ◽  
Cun-Hua Zhang
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Positive Equilibrium ◽  
Reaction Diffusion Model ◽  
Homogeneous Neumann Boundary Condition

A reaction–diffusion Gierer–Meinhardt system with homogeneous Neumann boundary condition on one-dimensional bounded spatial domain is considered in the present article. Local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are explored by analyzing in detail the associated eigenvalue problem. Moreover, properties of spatially homogeneous Hopf bifurcation are carried out by employing the normal form method and the center manifold technique for reaction–diffusion equations. Finally, numerical simulations are also provided in order to check the obtained theoretical conclusions.

Eigenvalues of −Δp − Δq Under Neumann Boundary Condition

2016 ◽  
Vol 59 (3) ◽  
pp. 606-616 ◽  
Author(s):  
Mihai Mihăilescu ◽  
Gheorghe Moroşanu
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Neumann Boundary Condition ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Careful Analysis ◽  
Full Description ◽  
Open Set ◽  
Homogeneous Neumann Boundary Condition ◽  
Isolated Point

AbstractThe eigenvalue problem −Δpu − Δqu = λ|u|q−2u with p ∊ (1,∞), q ∊ (2,∞), p ≠ q subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from ℝN with N ≥ 2. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval (λ1, ∞) plus an isolated point λ = 0. This comprehensive result is strongly related to our framework, which is complementary to the well-known case p = q ≠ 2 for which a full description of the set of eigenvalues is still unavailable.

Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains

2013 ◽  
Vol 14 (3) ◽  
pp. 819-850 ◽  
Author(s):  
Weizhu Bao ◽  
Qinglin Tang
Keyword(s):  
Boundary Condition ◽  
Numerical Methods ◽  
Neumann Boundary Condition ◽  
Landau Equation ◽  
Neumann Boundary ◽  
Quantized Vortex ◽  
Bounded Domains ◽  
Vortex Interaction ◽  
Ginzburg Landau ◽  
Homogeneous Neumann Boundary Condition

AbstractIn this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE) with a dimensionless parameter ε > 0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.

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