Nonexistence Of Stationary Solutions Research Articles

The role of Allee effects for Gaussian and Lévy dispersals in an environmental niche.

In the study of biological populations, the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. This effect supersedes the classical logistic model, in which low densities are favorable due to lack of competition, and includes situations related to deficit of genetic pools, inbreeding depression, mate limitations, unavailability of collaborative strategies due to lack of conspecifics, etc. The goal of this paper is to provide a detailed mathematical analysis of the Allee effect. After recalling the ordinary differential equation related to the Allee effect, we will consider the situation of a diffusive population. The dispersal of this population is quite general and can include the classical Brownian motion, as well as a Lévy flight pattern, and also a "mixed" situation in which some individuals perform classical random walks and others adopt Lévy flights (which is also a case observed in nature). We study the existence and nonexistence of stationary solutions, which are an indication of the survival chance of a population at the equilibrium. We also analyze the associated evolution problem, in view of monotonicity in time of the total population, energy consideration, and long-time asymptotics. Furthermore, we also consider the case of an "inverse" Allee effect, in which low density populations may access additional benefits.

On a free boundary model for three-dimensional MEMS with a hinged top plate: Stationary case

A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter \lambda which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of \lambda . Nonexistence of stationary solutions is obtained when \lambda is large enough.

Linear stability of the Vlasov–Poisson system with boundary conditions

We study the linear stability of the Vlasov–Poisson system in one dimension on a finite interval and half line with the in-flow and specular reflection boundary conditions. Conditions for the existence and non-existence of stationary solutions are obtained. The behavior of the perturbed solutions is investigated in different contexts. We show that, if the equilibrium distribution μ is a strictly decreasing function of local energy density, then the Vlasov–Poisson system on a finite interval is stable in the (weighted) L2 sense for the in-flow and specular reflection boundary conditions. For the case μ is strictly increasing, we prove a similar result on a finite interval under some extra assumptions. Finally, in the last part of the article, the spectral stability of the Vlasov–Poisson system on an unbounded interval is studied with similar monotonicity assumptions on μ.

The effect of shear in fiber spinning

AbstractWe study the equations of isothermal fiber spinning under the assumption that viscous friction in the fiber is balanced by shear stresses. Our discussion gives a rather complete picture of the existence and nonexistence of stationary solutions. The linearization about steady state of the governing equations is analyzed by semigroup methods and shown to have the spectrally determined growth property. Both linear and nonlinear stability of stationary solutions is investigated numerically.

Types and stability of detonation flows of aluminum particles in oxygen

The problem of detonation-wave structure is studied on the basis of a mathematical model for the detonation of aluminum particles in oxygen within the framework of a single-velocity two-temperature continuum. An analysis of flow types in the form of the Chapman-Jouguet strong-detonation and weak-detonation regimes is given. A chart of the mixture flow regimes in the plane of the Mach number of the detonation wave and the ratio of the characteristic times of thermal relaxation and combustion is constructed using the results of numerical experiments. The domain of realization of only strong detonation regimes, the manifold of existence of weak and strong detonation regimes, and the domain of nonexistence of stationary solutions are determined. The structural properties of the solutions with an internal singular point and weak structurally unstable regimes with a saddle singularity in the final state are described. The stability of all types of stationary regimes against small and finite perturbations that retain the detonation-wave (DW) velocity and the final state is shown by numerical modeling of nonstationary detonation flows.