Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics

In this paper we study a nonlocal reaction–diffusion–advection system modeling the growth of multiple competitive phytoplankton species in a vertical water column with incomplete mixing. We find that when the diffusion of the system is large, there is no positive steady states, and when the diffusion is not large, there exists at least one positive steady states under certain conditions. The main tools we use are the fixed point index theory, a refined comparison theorem and fine properties of the principal eigenvalues.     

Stability for steady states of Navier-Stokes-Poisson equations

In this paper, we study the stationary solution and nonlinear stability of Navier-Stokes-Poisson equations. Using variational method, we construct steady states of the N-S-P system as minimizers of a suitably defined energy functional, then show their dynamical stability against general, i.e. not necessarily spherically symmetric perturbation.     

Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation

We consider the Cauchy problem

     

On steady states in a collisionless plasma

Motivated by the collisionless shock problem in plasma, we study the stationary Vlasov-Maxwell system in one space dimension via a dynamical system approach. We construct flat-tail and oscillatory-tail solutions, as well as solitons in the presence of a nontrivial magnetic field. Flat-tail solutions connect different densities, pressures, and strengths of constant fields at x = ±∞, while oscillatory-tail solutions oscillate as x → −∞. We also construct topological horseshoes, which imply very complicated structures of some steady states. © 1996 John Wiley & Sons, Inc.. Inc.     

On the stability of steady states in a granuloma model

We consider a free boundary problem for a system of two semilinear parabolic equations. The system represents a simple model of granuloma, a collection of immune cells and bacteria filling a 3-dimensional domain Ω(t)$\Omega (t)$ which varies in time. We prove the existence of stationary spherical solutions and study their linear asymptotic stability as time increases to infinity.     

Asymptotic analysis of the steady stokes equation with randomly perturbed viscosity in a thin tube structure

The Stokes equation with the nonconstant viscosity is considered in a thin tube structure, i.e., in a connected union of thin rectangles with heights of order ε ≪ 1 and bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed. In the case of random perturbations of the constant viscosity, we prove that the leading term for the velocity is deterministic, while for the pressure it is random, but the expectations of the pressure satisfies the deterministic Darcy equation. Estimates for the difference between the exact solution and its asymptotic approximation are proved. Bibliography: 11 titles. Illustrations: 3 figures.     

Non-constant positive steady states of the Sel'kov model

This paper deals with the reaction-diffusion system known as the Sel'kov model with the homogeneous Neumann boundary condition. This model has been applied to various problems in chemistry and biology. We first give a priori estimates (positive upper and lower bounds) of positive steady states, and then study the non-existence, bifurcation and global existence of non-constant positive steady states as the parameters λ and θ are varied.     

一个趋化性模型非常数平衡解的存在性

对带两个趋化性参数的趋化性模型平衡解的存在性问题进行研究.在参数满足特定的条件下,应用局部分岔理论得到非常数平衡解的局部分岔结构,从而证明了该趋化性模型存在无穷多个非常数正平衡解.     

The steady states of a non-cooperative model of nuclear reactors

This paper characterizes the existence of coexistence states in a reaction-diffusion model arising in the theory of nuclear reactors. From a mathematical point of view, the importance of this model relies upon the fact that the associated variational systems are of non-cooperative type and, consequently, the comparison techniques available for cooperative systems fail to work out. Although in higher spatial dimensions the dynamics of the model might be rather involved, by the absence of limitations for the number of steady states, we can prove the uniqueness of the steady state in the one-dimensional prototype model. Our results complement and eventually sharpen the findings of Arioli [G. Arioli, Long term dynamics of a reaction-diffusion system, J. Differential Equations 235 (2007) 298-307].     

Non-constant positive steady states of a prey-predator system with cross-diffusions

In this paper, we study a strongly coupled elliptic system arising from a Lotka-Volterra prey-predator system, where cross-diffusions are included in such a way that the prey runs away from the predator and the predator moves away from a large group of preys. We establish the existence and non-existence of its non-constant positive solutions. Our results show that if m₁b<a<2m₁b/(1−m₁m₂) when 0<m₁m₂<1 or a>m₁b when m₁m₂?1, , d₂>0, d₃?0 and , then there exists (d₁,d₂,d₃,d₄) such that the stationary problem admits non-constant positive solutions. Otherwise, the stationary problem has no non-constant positive solution. In particular, the results indicate that its non-constant positive solutions are mainly created by the cross-diffusion d₄.     

Stability of the flow behind a pressure pulse in a tube

If a long tube with a slight overpressure is suddenly opened at one end, a pressure signal moves into the tube and a flow towards the open end is generated. The stability of this flow is investigated by solving the Orr-Sommerfeld equation for quasi-steady, onedimensional velocity profiles. Amplified disturbances are found for the boundary-layer type of flow close to the open end for a Reynolds number Re=u _mR/₀>17400 whereu _m=p ₀/ ₀ a ₀ is the maximum speed attained by the outflowing gas. A comparison of the time scales and of the amplification suggests that laminar flow can exist up to Re=100000.Experiments indicate that small disturbances due to the experiment can trigger transition at much lower Reynolds numbers.

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Zusammenfassung Ein langes Rohr mit einem schwachen Überdruck wird plötzlich an einem Ende geöffnet. Dadurch entsteht eine Druckwelle, die ins Rohr hinein läuft. Zwischen Druckwelle und Rohrende strömt das Fluid gegen das Rohrende. Die Stabilität dieser Strömung wird untersucht, indem die Orr-Sommerfeld Gleichung gelöst wird für quasistationäre und eindimensionale Geschwindigkeitsprofile. Die grenzschichtähnlichen Profile nahe dem Rohrende werden zuerst instabil für eine Reynoldszahl Re=u _mR/₀>17400 Dabei istu _m=p ₀/ ₀ a ₀ die maximale Geschwindigkeit, die das ausströmende Gas ereicht. Ein Vergleich von Zeitmaßstäben und Anfachung läßt erwarten, daß laminare Strömung bis gegen Re=100000 existieren kann.Versuche zeigen, daß kleinere Fehler im Versuchsaufbau den Umschlag bei viel kleineren Reynoldszahlen auslösen können.

     

Selection of steady states in planar Darcy convection

The planar natural convection of incompressible fluid in a porous medium is considered. We study selection of steady states under temperature perturbations on the boundary. A selection map is introduced to analyze the selection of steady state from a continuous family of equilibria which exists under zero boundary conditions. The results of finite-difference modeling for the rectangular enclosure are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of a steady flow guided by flexible walls

Flexible channels conveying fluid may loose stability by divergence or flutter, thus leading to undesirable dynamic behaviour. A lot of investigations have already been done on this subject, each involving structural models expressing a specific type of stiffness. The aim of this contribution is to investigate and compare systematically the influence of various types of structural stiffness on the stability behaviour of the coupled problem. Therefore, a simple, yet general model is built. The fluid is bounded by a rigid and an elastical supported, flexible wall, which exhibits bending and extensional stiffness and considers effects of pre-load. The influence of the various characteristics on the stability of the steady state is discussed and a perturbation approach is used to investigate the influence of small nonlinearities. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of steady rotations in the nonholonomic Routh problem

We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.      

Laminar film condensation in a vertical tube in the presence of noncondensable gas

A two-dimensional, steady state model of convective film-wise condensation of a vapor and noncondensable gas mixture flowing downward inside a vertical tube is developed. The mass transfer at the film and gas interface is treated as diffusion controlled process. The noncondensable effect on the condensation is taken into account through boundary layer analysis of species concentration and energy balance. Numerical predictions were obtained for the condensation heat transfer coefficient of turbulent vapor flow associated with laminar condensate. The predictions were compared with the experimental data in the literature to assess the model. Noncondensable mass fraction and vapor–noncondensable mixture temperature were presented in the form of radial and axial profiles.     

Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion

This paper deals with weak solutions of the one-dimensional viscous Hamilton-Jacobi equation

     

On the stability of the positive radial steady states for a semilinear Cauchy problem

This paper is a contribution to the analysis of the Cauchy problem (0.1)$\{\begin{array}{c}\frac{\partial u}{\partial t}=\Delta u+\sum _{i=1}^k{c}_i{\left|x\right|}^{l_i}{u}^{p_i}\text{in}{\mathbb{R}}^n\times (0,T)\text{,}\hfill \\ u(x,0)=\phi \left(x\right)\text{in}{\mathbb{R}}^n\text{,}\hfill \end{array}$ with initial function $\phi \not\equiv 0$. The stability and asymptotic stability of the positive radial steady states, which are positive solutions of (0.2)$\Delta u+\sum _{i=1}^k{c}_i{\left|x\right|}^{l_i}{u}^{p_i}=0\text{,}$ have been discussed with $\phi$ in weighted Banach spaces.