Existence and uniqueness of maximal strong solution of a 1D blood flow in a network of vessels

We study the well-posedness of a system of one-dimensional partial differential equations modeling blood flows in a network of vessels with viscoelastic walls. We prove the existence and uniqueness of maximal strong solution for this type of hyperbolic/parabolic model. We also prove a stability estimate under suitable nonlinear Robin boundary conditions.     

On a Problem of Ulam

We prove stability of solutions to one minimization problem for an integral functional.     

The existence of a periodic solution in a tide dynamics problem

We examine a quasilinear boundary-value problem describing the dynamics of tidal flow in the sea. For small values of ground friction coefficient and under the fulfillment of a certain condition on the function giving the depth of the sea, we prove the existence of a generalized periodic solution. We construct a difference scheme for the numerical solution of the problem being examined and we prove its stability.Translated from Problemy Matematicheskogo Analiza. No. 4: Integralnye i Differentsial'nye Operatory. Differentsial'nye Uraveniya, pp. 3–9, 1973.     

On stability of a family of functional equations

We prove a stability result for a family of functional equations containing the exponential and the Go??b-Schinzel functional equations. Thus we extend and improve a recent result of J. Chudziak.     

一类多因素HIV模型的研究

研究了一类多因素HIV模型.建立了一个标准型的DI模型,证明了无病平衡点的局部渐近稳定性和全局渐近稳定性.     

Stability Analysis and Attractivity Results of a DC-DC Buck Converter

Using tools from set-valued and variational analysis, we propose a mathematical formulation for a power DC-DC Buck converter. We prove the existence of trajectories for the model. A stability and asymptotic stability results are established. The theoretical results are supported by some numerical simulations with a discussion about explicit and implicit schemes.     

Dynamics of a rate equation describing cluster-size evolution

In this paper, we show dynamics of Smoluchowski's rate equation which has been widely applied to studies of aggregation processes (i.e., the evolution of cluster-size distribution) in physics. We introduce dissociation in the rate equation while dissociation is neglected in previous works. We prove the positiveness of solutions of the equation, which is a basic guarantee for the effectiveness of the model since the possibility that some solution may be negative is excluded. For the case of cluster coalesce without dissociation, we show both the equilibrium uniqueness and the equilibrium stability under the condition that the monomer deposition stops. For the case that clusters evolve with dissociation and there is no monomer deposition, we show the equilibrium uniqueness and prove the equilibrium stability if the maximum cluster size is not larger than three while we show the equilibrium stability by numerical simulations if the maximum size is larger than three.     

一类高阶有理型差分方程的定性研究

主要研究一类高阶有理差分方程的定性问题,证明了方程正解的渐近稳定性、有界性、周期性以及全局吸引性.     

Persistence and propagation of a discrete-time map and PDE hybrid model with strong Allee effect

Persistence and propagation of species are fundamental questions in spatial ecology. This paper focuses on the impact of Allee effect on the persistence and propagation of a population with birth pulse. We investigate the threshold dynamics of an impulsive reaction–diffusion model and provide the existence of bistable traveling waves connecting two stable equilibria. To prove the existence of bistable waves, we extend the method of monotone semiflows to impulsive reaction–diffusion systems. We use the methods of upper and lower solutions and the convergence theorem for monotone semiflows to prove the global stability of traveling waves and their uniqueness up to translation. Then we enhance the stability of bistable traveling waves to global exponential stability. Numerical simulations illustrate our theoretical results.     

Stability of static walls for a three-dimensional model of ferromagnetic material

In this paper we consider a three-dimensional model of ferromagnetic material. We deal with the static domain wall configuration calculated by Walker. We prove the stability of this configuration for the Landau–Lifschitz equation with a simplified expression of the demagnetizing field.     

Lyapunov exponent of a stochastic SIRS model

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.     

Stability of a linear difference equation and estimation of its fundamental solution

We consider a linear nonautonomous system of difference equations. Under natural restrictions we prove that the stability with respect to the initial function and that with respect to the right-hand side in spaces of bounded and summable functions are equivalent to the corresponding estimates for the fundamental solution. As a consequence, we obtain an effective stability condition for a system with a constant matrix.     

The existence and stability of steady states for a prey-predator system with cross diffusion of quasilinear fractional type

This paper is concerned with the existence and stability of steady states for a prey-predator system with cross diffusion of quasilineax fractional type. We obtain a sufficient condition for the existence of positive steady state solutions by applying bifurcation theory and a detailed priori estimate. In virtue of the principle of exchange of stability, we prove the stability of local bifurcating solutions near the bifurcation point.     

The Existence and Stability of Steady States for a Prey-predator System with Cross Difusion of Quasilinear Fractional Type

This paper is concerned with the existence and stability of steady states for a prey-predator system with cross difusion of quasilinear fractional type.We obtain a sufcient condition for the existence of positive steady state solutions by applying bifurcation theory and a detailed priori estimate.In virtue of the principle of exchange of stability,we prove the stability of local bifurcating solutions near the bifurcation point.     

The dirichlet problem for a loaded mixed-type equation in a rectangular domain

We establish necessary and sufficient conditions for the unique solvability of the first boundary problem for a loaded equation with the Lavrent’ev-Bitsadze operator in a rectangular domain. We obtain a solution to the stated problem as the sum of the eigenfunction series for the corresponding one-dimensional problem with respect to eigenvalues. We prove the stability of the solution with respect to boundary functions.     

Inverse problem for an equation of parabolic-hyperbolic type with a nonlocal boundary condition

For an equation of the parabolic-hyperbolic type, we consider an inverse problem with a nonlocal condition relating solution derivatives that belong to different types of the equation in question. We justify a uniqueness criterion and prove the existence of a solution of the problem by the spectral analysis method. We prove the stability of the solution with respect to the nonlocal boundary condition.     

The stability radius of a semi-Fredholm operator

We prove an asymptotic formula for the stability radius of a semi-Fredholm operator on a Banach space in terms of the reduced minimum modulus. In particular, this gives a new proof of the Fredholm case studied by Förster and Kaashoek [4].     

Stabilization of a transmission wave/plate equation

We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchhoff plate equation. We prove an exponential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate.     

关于一类多项式族的鲁棒稳定性研究

本文对包括区间多项式族和菱形多项式族的一类多项式族的鲁棒稳定性进行了研究.我们给出并证明了其中几个可用有限检验来判断的多项式族的Hurwitz稳定的实例;同时举例说明了有限检验对所有这一类多项式族并不总是可行的.     

Stability with respect to the input data and monotonicity of an implicit finite-difference scheme for a quasilinear parabolic equation

We analyze the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a quasilinear parabolic equation under specific conditions imposed solely on the problem input data. We prove some kinds of the maximum principle for the nonlinear equations that are used in the derivation of a priori estimates for the solution; we also prove estimates for some kinds of recursion inequalities that are used in the derivation of a priori estimates for higher-order derivatives, these estimates being necessary for proving the continuous dependence of the solution on small perturbations of the input data and for analyzing monotonicity in the nonlinear case. We show that, depending on the properties of the input data, higher derivatives can become infinite in finite critical time. We obtain conditions on the input data guaranteeing the stability of the difference scheme on the entire time interval.