Asymptotic Stability Of Stationary Solution Research Articles

Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space

We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝn admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted L p energy method.

Stability of stationary solutions of piecewise affine differential equations describing gene regulatory networks

We study stability properties of a class of piecewise affine systems of ordinary differential equations arising in the modeling of gene regulatory networks. Our method goes back to the concept of a Filippov stationary solution (in the narrow sense) to a differential inclusion corresponding to the system in question. The main result of the paper justifies a reduction principle in the stability analysis enabling to omit the variables that are not singular, i.e. that stay away from the discontinuity set of the system. We suggest also “the first approximation method” to study asymptotic stability of stationary solutions based on calculating the principal part of the system, which is 0-homogeneous rather than linear. This leads to an efficient algorithm of how to check asymptotic stability without calculating the eigenvalues of the systemʼs Jacobian. In Appendix A we discuss and compare two other concepts of stationary solutions to the system in question.

Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics

The main concern of the present paper is to analyze a sheath formed around a surface of a material with which plasma contacts. Here, for a formation of the sheath, the Bohm criterion requires the velocity of positive ions should be faster than a certain physical constant. The behavior of positive ions in plasma is governed by the Euler-Poisson equations. Mathematically, the sheath is regarded as a special stationary solution. We first show that the Bohm criterion gives a sufficient condition for an existence of the stationary solution by using the phase plane analysis. Then it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in the weighted Sobolev space. Moreover we obtain the convergence rate of the time global solution towards the stationary solution subject to the decay rate of the initial perturbation. These theorems are proved by a weighted energy method.

Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues

This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration $\sigma$ satisfies the stationary diffusion equation $\Delta\sigma=f(\sigma)$ and proved that there exists a threshold value $\gamma_*>0$ for the surface tension coefficient $\gamma$, such that the radial stationary solution is asymptotically stable in case $\gamma>\gamma_*$, while unstable in case $\gamma<\gamma_*$. In this paper we extend this result to the case where $\sigma$ satisfies the nonstationary diffusion equation $\varepsilon\partial_t\sigma=\Delta\sigma-f(\sigma)$. We prove that for the same threshold value $\gamma_*$ as above, for every $\gamma>\gamma_*$, there is a corresponding constant $\varepsilon_0(\gamma)>0$ such that for any $0<\varepsilon<\varepsilon_0(\gamma)$, the radial stationary solution is asymptotically stable with respect to small enough nonradial perturbations, while for $0<\gamma<\gamma_*$ and $\varepsilon$ sufficiently small, it is unstable under nonradial perturbations.

A Parabolic–Hyperbolic Quasilinear System

We prove the existence and uniqueness of solutions of the parabolic–hyperbolic system for unknown (u,φ) = (u(x,t),φ(x,t)), where Ω is a bounded domain in ℝ n with smooth boundary. The system is solved subject to no-flux boundary condition for φ and given initial conditions for u and φ. Structural conditions on F and G are assumed to ensure L ∞ a priori estimates on u and φ. The key for global in time existence is the Lipschitz continuity of ∇φ in the spatial variable x, an intrinsic requirement for the existence of classical solutions to the hyperbolic equation u t + ∇φ · ∇ u = Q. A new method, worked specifically for hyperbolic-parabolic or hyperbolic-elliptic systems, is developed here to establish an L ∞ bound for the Hessian . In the final part of the paper we prove the asymptotic stability of stationary solutions.

Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors

In this paper we study well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors under the action of external inhibitors. An important feature of this problem is that the surface tension of the free boundary is taken into account. We first reduce this free boundary problem into an evolution equation in little Hölder space and use the well-posedness theory for differential equations in Banach spaces of parabolic type (i.e., equations which are treatable by using the analytic semi-group theory) to prove that this free boundary problem is locally well-posed for initial data belonging to a little Hölder space. Next we study flat solutions of this problem. We obtain all flat stationary solutions and give a precise description of asymptotic stability of these stationary solutions under flat perturbations. Finally we investigate asymptotic stability of flat stationary solutions under non-flat perturbations. By carefully analyzing the spectrum of the linearized stationary problem and employing the theory of linearized stability for differential equations in Banach spaces of parabolic type, we give a complete analysis of stability and instability of all flat stationary solutions under small non-flat perturbations.

Pseudospectra for matrix pencils and stability of equilibria

The concept of e-pseudospectra for matrices, introduced by Trefethen and his coworkers, has been studied extensively since 1990. In this paper, e-pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of e-pseudospectra for matrix pencils will be discussed. As an application, we demonstrate how this concept can be used for investigating the asymptotic stability of stationary solutions to time-dependent ordinary or partial differential equations; two cases, based on Burgers' equation, will be shown.

On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations

We consider nonlinear wave and Klein-Gordon equations with general nonlinear terms, localized in space. Conditions are found which provide asymptotic stability of stationary solutions in local energy norms. These conditions are formulated in terms of spectral properties of the Schrodinger operator corresponding to the linearized problem. They are natural extensions to partial differential equations of the known Lyapunov condition. For the nonlinear wave equation in three-dimensional space we find asymptotic expansions, as t→∞, of the solutions which are close enough to a stationary asymptotically stable solution.

On the stability behaviour of systems obtained by index-reduction

Index-reduction may destabilize the asymptotic stability of stationary solutions. We analyze in some detail how index-reduction via an artificial Lagrangian multiplier as well as the Baumgarte method behaves.

Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation

AbstractWe study a simplified model of barotropic flow. The governing equation is the two‐dimensional Euler equation for incompressible flow, subject to Ekman dissipation and arbitrary forcing, in a bounded domain. This equation is converted to an equivalent functional differential equation via a semi‐lagrangian formalism. We use Hölder Cα estimates and compactness arguments for the latter equation to deduce existence, uniqueness and asymptotic stability of stationary solutions under the assumption of small forcing with respect to the dissipation constant.In case the forcing is of one sign only throughout the domain, we are able to deduce existence without any further assumptions. The same result holds in the limit of the dissipation constant approaching zero while the ratio of this constant to the forcing is fixed. Thus we get the existence of special types of stationary solutions to the inviscid Euler equation consistent with the given forcing.