Existence Of Steady-state Solutions Research Articles

Hopf bifurcation in a Lotka-Volterra competition-diffusion-advection model with time delay

In this paper, we mainly investigate a Lotka-Volterra competition-diffusion-advection system with time delay, where the diffusion and advection rates of two competitors are different. By employing the Lyapunov-Schmidt reduction method, we obtain the existence of steady state solution. A weighted inner product has been introduced to study stability and Hopf bifurcation at the spatially nonhomogeneous steady-state. Our results imply that the infinitesimal generator associated with the linearized system have two pairs of purely imaginary eigenvalues, and time delay can make the spatially nonconstant positive steady state unstable for a reaction-diffusion-advection model. In addition, the bifurcation direction and stability of Hopf bifurcating periodic orbits was obtained by means of the center manifold reduction and the normal form theory.

Stability and bifurcation in a two-species reaction–diffusion–advection competition model with time delay

In this paper, we are concerned with the dynamics of a class of two-species reaction–diffusion–advection competition models with time delay subject to the homogeneous Dirichlet boundary condition or no-flux boundary condition in a bounded domain. The existence of steady state solution is investigated by means of the Lyapunov–Schmidt reduction method. The stability and Hopf bifurcation at the spatially nonhomogeneous steady-state are obtained by analyzing the distribution of the associated eigenvalues. Finally, the effect of advection on Hopf bifurcation is explored, which shows that with the increase of convection rate, the Hopf bifurcation phenomenon is more likely to emerge.

Propagation dynamics for monotone evolution systems without spatial translation invariance

In this paper, under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of monotone evolution systems without spatial translation invariance. Then we apply the developed theory to study traveling waves and spatio-temporal propagation patterns for time-delayed nonlocal equations, reaction-diffusion equations in a cylinder, and asymptotically homogeneous KPP-type equations. We also obtain the existence of steady state solutions and asymptotic spreading properties of solutions for a time-delayed reaction-diffusion equation subject to the Dirichlet boundary condition.

Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation.

In this paper we consider a time-delayed mathematical model describing tumor growth with angiogenesis and Gibbs-Thomson relation. In the model there are two unknown functions: One is $\sigma(r,t)$ which is the nutrient concentration at time $t$ and radius $r$, and the other one is $R(t)$ which is the outer tumor radius at time $t$. Since $R(t)$ is unknown and varies with time, this problem has a free boundary. Assume $\alpha(t)$ is the rate at which the tumor attracts blood vessels and the Gibbs-Thomson relation is considered for the concentration of nutrient at outer boundary of the tumor, so that on the outer boundary, the condition $$\dfrac{\partial \sigma}{\partial r}+\alpha(t)\left(\sigma-N(t)\right)=0,~~r=R(t)$$ holds, where $N(t)=\bar{\sigma}\left(1-\dfrac{\gamma}{R(t)}\right)H(R(t))$ is derived from Gibbs-Thomson relation. $H(\cdot)$ is smooth on $(0,\infty)$ satisfying $H(x)=0$ if $x\leq \gamma$, $H(x)=1$ if $x\geq 2\gamma$ and $0\leq H'(x)\leq 2/\gamma$ for all $x\geq 0$. In the case where $\alpha$ is a constant, the existence of steady-state solutions is discussed and the stability of the steady-state solutions is proved. In another case where $\alpha$ depends on time, we show that $R(t)$ will be also bounded if $\alpha(t)$ is bounded and some sufficient conditions for the disappearance of tumors are given.

Shear-density coupling for a compressible single-component yield-stress fluid.

Flow behavior of a single-component yield stress fluid is addressed on the hydrodynamic level. A basic ingredient of the model is a coupling between fluctuations of density and velocity gradient via a Herschel-Bulkley-type constitutive model. Focusing on the limit of low shear rates and high densities, the model approximates well-but is not limited to-gently sheared hard sphere colloidal glasses, where solvent effects are negligible. A detailed analysis of the linearized hydrodynamic equations for fluctuations and the resulting cubic dispersion relation reveals the existence of a range of densities and shear rates with growing flow heterogeneity. In this regime, after an initial transient, the velocity and density fields monotonically reach a spatially inhomogeneous stationary profile, where regions of high shear rate and low density coexist with regions of low shear rate and high density. The steady state is thus maintained by a competition between shear-induced enhancement of density inhomogeneities and relaxation via overdamped sound waves. An analysis of the mechanical equilibrium condition provides a criterion for the existence of steady state solutions. The dynamical evolution of the system is discussed in detail for various boundary conditions, imposing either a constant velocity, shear rate, or stress at the walls.

Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management

A reaction-diffusion logistic population model with spatially nonhomogeneous harvesting is considered. It is shown that when the intrinsic growth rate is larger than the principal eigenvalue of the protection zone, then the population is always sustainable; while in the opposite case, there exists a maximum allowable catch to avoid the population extinction. The existence of steady state solutions is also studied for both cases. The existence of an optimal harvesting pattern is also shown, and theoretical results are complemented by some numerical simulations for one-dimensional domains.

Stability analysis of a stage structure model with spatiotemporal delay effect

This paper is concerned with a stage structure model with spatiotemporal delay and homogeneous Dirichlet boundary condition. The existence of steady state solution bifurcating from the trivial equilibrium is obtained by using Lyapunov–Schmidt reduction. The stability analysis of the positive spatially nonhomogeneous steady state solution is investigated by a detailed analysis of the characteristic equation. Using the properties of the omega limit set, we obtain the global convergence of the solution with finite delay.

Steady-state and Hopf bifurcations in the Langford ODE and PDE systems

This paper is concerned with the Langford ODE and PDE systems. For the Langford ODE system, the existence of steady-state solutions is firstly obtained by Lyapunov–Schmidt method, and the stability and bifurcation direction of periodic solutions are established. Then for the Langford PDE system, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalue. Finally, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions for the PDE system are investigated.

Effects of temperature dependence of electrical and thermal conductivities on the Joule heating of a one dimensional conductor

We examine the effects of temperature dependence of the electrical and thermal conductivities on Joule heating of a one-dimensional conductor by solving the coupled non-linear steady state electrical and thermal conduction equations. The spatial temperature distribution and the maximum temperature and its location within the conductor are evaluated for four cases: (i) constant electrical conductivity and linear temperature dependence of thermal conductivity, (ii) linear temperature dependence of both electrical and thermal conductivities, (iii) the Wiedemann–Franz relation for metals, and (iv) polynomial fits to measured data for carbon nanotube fibers and for copper. For (i) and (ii), it is found that there are conditions under which no steady state solution exists, which may indicate the possibility of thermal runaway. For (i), analytical solutions are constructed, from which explicit expressions for the parameter bounds for the existence of steady state solutions are obtained. The shifting of these bounds due to the introduction of linear temperature dependence of electrical conductivity (case (ii)) is studied numerically. These results may provide guidance in the design of circuits and devices in which the effects of coupled thermal and electrical conduction are important.

Dynamics of a delayed diffusive predator–prey model with hyperbolic mortality

This paper is devoted to consider a time-delayed diffusive prey–predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction–diffusion model without time delay.

Effect of atoms evaporated from the anode on the structure of the vacuum arc space charge sheath

The structure of the anode space charge sheath of a vacuum arc is studied with allowance for the dependence of the negative anode fall on the ratio of the directed electron velocity v 0 to the electron thermal velocity v T for different values of the flux density of atoms evaporated from the anode. Poisson’s equation for the sheath potential is solved taking into account the electron space charge, fast cathode ions, and slow ions produced due to the ionization of atoms evaporated from the anode. The kinetic equation for atoms and slow anode ions is solved with allowance for ionization in the collision integral. Analytic solutions for the velocity distribution functions of atoms and slow ions and the density of slow ions are obtained. It is shown that the flux of slow ions substantially affects the spatial distribution of the electric field E(z) in the sheath. As the flux density increases, the nonmonotonic dependence E(z) transforms into a monotonic one and the sheath narrows. For a given flux of evaporated atoms Πa, the increase in the ratio of the directed electron velocity to the electron thermal velocity leads again to a nonmonotonic dependence E(z). As z increases, the electric field first increases, passes through the maximum, decreases, passes through the minimum E min, and then again increases toward the anode. There is a limiting value of the ratio (v 0/v T )* at which E min(z) vanishes. At v 0/v T > (v 0/V T )*, the condition for the existence of a steady-state sheath is violated and the profiles of the field and potential in the sheath become oscillating. The dependence of (v 0/v T )* on the flux density of evaporated atoms Π a is obtained. It is shown that the domain of existence of steady-state solutions in the sheath broadens with increasing Π a .

Existence of Steady State Solutions with Finite Energy for the Magnetohydrodynamic Equations in the Whole Space R3

We study the steady state Magnetohydrodynamic (MHD) equations in the whole space Following the work of C. Bjorland and M. Schonbek (4) onNavier -Stokes equations in the whole space, we prove the existence of atleast one solution with finite Dirichlet Integral to steady state Magnetohydrodynamic equations in the whole space. Further, we show that these solutions are unique among all solutions with finite energy and finite DirichletIntegral . I. Introduction Magnetohydrodynamics (MHD) is the study of flows of fluids which are electrically conducting and move in a magnetic field. The simplest example of an electrically conducting fluid is a liquid metal like mercury or liquid sodium. MHD treats, in particular, conducting fluids either in liquid form or gaseous form.The equations describing the motion of a viscous incompressible conducting fluid moving in a magnetic field are derived by coupling Navier-Stokes equations with Maxwell's equations together with expression for the Lorentz force. The domain Ω in which the fluid is moving is either a bounded subset of or the whole space . In this paper we restrict our considerations to a domain Ω which is the whole space .During past four or five decades, there have been an extensive study of qualitativeproperties such as existence, uniqueness, regularity and stability of solutions of theMHD equations. This is evident from the work of Duvaut and Lions (1), E. SanchezPalencia (2), Sermange and Temam (3) and other researchers working in the field. Themethods from nonlinear functional analysis such as Galerkin approximation, fixed pointtheorems, monotone and coercive operators, semigroup theory etc have been applied toestablish many a qualitative properties for compressible as well as incompressible MHDflows. The function spaces used are either Holder spaces or Sobolev spaces which are theappropriate function spaces for using these methods and the theory of elliptic operators. In spite of these works, there are very few qualitative results available in the case where the domain is the full space. In the case when domain is a bounded subset of R 3 , it is easy to obtain qualitative results by using Poincare type inequality. But for unbounded domain, one has to use other techniques as were developed by C. Bjorland and M. Schonbek (4). As for MHD flows for incompressible conducting fluids, there are other works where regularity results for MHD flows have been proved ( see references (5-7) ). However, as in the case of Navier-Stokes equations for incompressible fluids, the proof of global regularity remains illusive in this case also. In the present paper, we show that the techniques used in (4) can be extended to prove similar results for steady state Magnetohydrodynamic (MHD) flows. Thus, we consider viscous incompressible Magnetohydrodynamic (MHD) flow governed by the following equations:

Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies

Robust morphogen gradient formation is important for embryo development. Patterns of developmental tissue are encoded by the morphogen gradient that drives the process of cell differentiation in response to different morphogen levels. Experiments have shown that tissue patterning is robust with respect to morphogen overexpression. However, the mechanisms for this robust patterning remain unclear. The expansion-repression mechanism, which was proposed for achieving scaling of patterning with organ size, is a type of self-enhanced clearance through a non-local feedback regulation and may contribute to the robustness with respect to morphogen overexpression. In this paper, we study the role of the expansion-repression mechanism in morphogen gradient robustness through a two-equation model with general forms of feedback functions. We prove the existence of steady-state solutions, and, through model reduction and simplification, show that the expansion-repression mechanism is able to improve the robustness against changes in the morphogen production rate. However, this improvement is restricted by the biological requirement of multi-fate long-range morphogen gradient.

Photonic lattices

We use critical point theory to find periodic solutions of the nonlinear steady state Schrödinger equations arising in the study of photonic lattices. We show that nontrivial solutions exist for wide ranges of the parameters. It follows that there is a large continuous energy or wavenumber spectrum that allows the existence of steady state solutions. Our results hold in arbitrary dimensions.

Steady State Solution to Atmospheric Circulation Equations with Humidity Effect

The steady state solution to atmospheric circulation equations with humidity effect is studied. A sufficient condition of existence of steady state solution to atmospheric circulation equations is obtained, and regularity of steady state solution is verified.

Existence of steady-state solutions in a nonlinear photonic lattice model

Many careful experimental observations to nonlinear photonic lattice model have been constructed. In this paper, we use the principle of variational method, mountain pass lemma, fixed point method to develop an existence theorem for the steady-state solutions of a nonlinear photonic lattice model describing the propagation of a light wave in a photo-refractive crystal is established, which demonstrates that there is an amount of continuous energy that allows the existence of steady-state solutions. Our results provide a theoretical principles for a variety of experiments and research on photonic lattices and crystals. Finally, it is straightforward to see that the applicability of the present to constructing arbitrarily small energy solutions is also guaranteed.

Steady State Solution and Stability of an Age-Structured MSIQR Epidemic Model

The importance of epidemiology in our life has stimulated researchers to extend the classic Susceptibles-Infectives-Removed (SIR) model to sophisticated models by including more factors in order to give detailed transmission dynamics of epidemic diseases. However, the integration of the quarantine policy and age-structure is less addressed. In this work we propose an age-structured MSIQR (temporarily immune-susceptibles-infectives-quarantined-removed) model to study the 0 1   impact of quarantine policies on the spread of epidemic diseases. Specifically, we investigate the existence of steady state solutions and stability property of the proposed model. The derived explicit expression of the basic reproductive number shows that the disease-free equilibrium is globally asymptotically stable if, and that the unique endemic equilibrium exists if. In addition, the stability conditions of the endemic equilibrium are derived.

Flat Galaxies with Dark Matter Halos—Existence and Stability

We consider a model for a flat, disk-like galaxy surrounded by a halo of dark matter, namely a Vlasov-Poisson type system with two particle species, the stars which are restricted to the galactic plane and the dark matter particles. These constituents interact only through the gravitational potential which stars and dark matter create collectively. Using a variational approach we prove the existence of steady state solutions and their nonlinear stability under suitably restricted perturbations.

Quantum electron-acoustic double layers in a magnetoplasma

Using a quantum magnetohydrodynamic (QMHD) model, the existence of small but finite amplitude quantum electron-acoustic double layers (QEADLs) is reported in a magnetized collisionless dense quantum plasma whose constituents are two distinct groups of cold and hot electrons, and the stationary ions forming only the neutralizing background. It is shown that the existence of steady state solutions of these double layers obtained from an extended Korteweg-de Vries (KdV) equation depends parametrically on the ratio of the cold to hot electron unperturbed number density (δ), the quantum diffraction parameter (H), the obliqueness parameter (lz), and the external magnetic field via the normalized electron-cyclotron frequency (Ω). It is found that the system supports both compressive and rarefactive double layers depending on the parameters δ and lz. The effects of all these parameters on the profiles of the double layers are also examined numerically.

Existence of steady freefall of rigid bodies in a second order fluid with applications to particle sedimentation

We study the slow motion of rigid bodies of arbitrary shape sedimenting in a quiescent viscoelastic liquid under the action of gravity. The liquid is modeled by the full second order fluid equations with arbitrary material parameters α 1 + α 2 . We show existence of steady state solutions for small Weissenberg numbers and zero Reynolds numbers and apply our equations to study the steady state orientations of symmetric bodies sedimenting in viscoelastic fluids.