On Second-Order Differentiability with Respect to Parameters for Differential Equations with State-Dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show first and second-order differentiability of the solution with respect to parameters in a pointwise sense and also using the $C$ -norm on the state-space, assuming that the state-dependent time lag function is piecewise strictly monotone.     

Regularization of Neutral Delay Differential Equations with Several Delays

For neutral delay differential equations the right-hand side can be multi-valued, when one or several delayed arguments cross a breaking point. This article studies a regularization via a singularly perturbed problem, which smooths the vector field and removes the discontinuities in the derivative of the solution. A low-dimensional dynamical system is presented, which characterizes the kind of generalized solution that is approximated. For the case that the solution of the regularized problem has high frequency oscillations around a codimension-2 weak solution of the original problem, a new stabilizing regularization is proposed and analyzed.     

Global Stability of a Competitive Model with State-Dependent Delay

This article deals with a stage-structured model with state-dependent delay which is assumed to be an increasing function of the population density with lower and upper bound. Firstly, according to the principle of linearized stability (Theorem 3.6, Hartung et al. in Handbook of differential equations: ordinary differential equations, 2006), we study the local stability of system in combination with the positivity and boundedness of solutions. By using the comparison principle obtained and an iterative method, the global stability of the equilibria is completely analyzed. Our results show how the interaction between interspecific and intraspecific competition affects the coexistence of both species.     

四神经元时滞网络的稳定性与分岔

本文研究具有长连接的四神经元构成的时滞网络的稳定性与分叉.网络系统可采用一组含有多个时滞的非线性微分方程描述.通过分析网络零平衡点处线性近似系统特征方程的根的分布情况,给出了网络零平衡点全时滞局部渐近稳定条件和与时滞相关的局部渐近稳定条件.讨论网络平衡点的数目及稳定性,得到了网络静态分叉产生条件.以时滞为分叉参数,给出了零平衡点失稳后网络出现由Hopf分叉引起的周期运动的存在性条件.分叉周期运动的性质由网络的非线性因素确定.借助中心流形定理和规范型理论,得到了确定Hopf分叉方向,分叉周期运动稳定性和分叉周期运动周期的公式.最后给出数值算例,验证了理论分析的结果.研究表明:网络中长连接的强度和性质对网络的动力学行为有着重要的影响.     

Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRⁿ\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.     

Lyapunov Stability and Adaptive Regulation of a Class of Nonlinear Nonautonomous Second-Order Differential Equations

This paper presents an adaptive regulation scheme for a class ofordinary nonlinear nonautonomous second-order differential equationswhich includes as particular cases a number of particular differentialequations which occur in applications. The unforced reference model isproposed to be a stable differential parametrization within the generalclass dealt with. Therefore, some sufficient Lyapunov's stabilityconditions for such a class are previously investigated which can beused, in particular, to set an appropriate reference model. Theresulting closed-loop adaptive scheme is proved to be stable and itinvolves a parameter estimation scheme of least-squares type which isproved to possess all suitable properties in terms of estimatesboundedness and asymptotic convergence of the estimates to finite limitsas well as time-integrability of the squared adaptation error.     

Global Regularity for a Class of Generalized Magnetohydrodynamic Equations

It remains unknown whether or not smooth solutions of the 3D incompressible MHD equations can develop finite-time singularities. One major difficulty is due to the fact that the dissipation given by the Laplacian operator is insufficient to control the nonlinearity and for this reason the 3D MHD equations are sometimes regarded as “supercritical”. This paper presents a global regularity result for the generalized MHD equations with a class of hyperdissipation. This result is inspired by a recent work of Terence Tao on a generalized Navier–Stokes equations (T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equations, arXiv: 0906.3070v3 [math.AP] 20 June 2009), but the result for the MHD equations is not completely parallel to that for the Navier–Stokes equations. Besov space techniques are employed to establish the result for the MHD equations.     

Differentiability of Solutions with Respect to the Initial Data in Differential Equations with State-dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is strictly monotone increasing.     

Regularity Results for Solutions of a Class of Hamilton-Jacobi Equations

The regularity of the gradient of viscosity solutions of first‐order Hamilton‐Jacobi equations is studied under a strict convexity assumption on H(t,x,⋅). Estimates on the discontinuity set of Du are derived. Such estimates imply that solutions of the above problem are smooth in the complement of a closed ℋ ⁿ ‐rectifiable set. In particular, it follows that Du belongs to the classSBV, i.e., D ² u$ is a measure with no Cantor part. (Accepted February 12, 1996)     

Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications

The paper concerns a class of n-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.     

Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations

Initial value problems for quasilinear parabolic equations having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon measures of the initial data. We call such solutions measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the measure-valued solution of problem (P) is, in fact, function-valued.     

Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable,Hyperbolic Case

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}(t)&=f(\theta ,x(t),\epsilon x(t-\tau (x(t))))\\ \dot{\theta }(t)&=\omega . \end{aligned} \right. \end{aligned}$$

Under the assumption of exponential dichotomies for the \(\epsilon =0\) case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc.

     

Pseudo Almost Periodic Solutions to a Class of Semilinear Differential Equations

This paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear differential equations involving the algebraic sum of two (possibly noncommuting) densely defined closed linear operators acting on a Hilbert space. Sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those semilinear equations are obtained. An erratum to this article is available at .     

Evolution Equations for the Surface Concentration of an Insoluble Surfactant; Applications to the Stability of an Elongating Thread and a Stretched Interface

The general form of the convection–diffusion equation governing the evolution of the surface concentration of an insoluble surfactant over an evolving interface is reviewed and discussed for three-dimensional, axisymmetric, and two-dimensional configurations. The linearized form of the evolution equation is then derived around cylindrical and planar shapes in a framework that is suitable for carrying out a linear stability analysis for axisymmetric or two-dimensional perturbations. Particular attention is paid to the cases of quiescent unperturbed fluids, unidirectional shear flow, and elongational flow. By way of application, the linearized transport equations are combined with Stokes-flow hydrodynamics to investigate the stability of an elongating cylindrical viscous thread suspended in an ambient viscous fluid or in a vacuum, and the stability of a two-dimensional interface separating two semi-infinite fluids and stretched under the action of an orthogonal stagnation-point flow. The results illustrate the possibly important role of the surfactant on the linear growth of periodic waves on the cylindrical interface, and reveal that the surfactant has no effect on the stability of the planar interface.     

On the Structure of the Set of Solutions for One Class of Systems of Nonlinear Differential-Functional Equations of Neutral Type

For one class of nonlinear differential-functional equations, we study the structure of the set of its solutions continuously differentiable for t R ⁺ = [0, +).     

Symbolic Sequences Generated by a Special Class of Discrete Systems With Discontinuity and Input

In this paper, we study the symbolic sequences generated by aclass of discrete systems. This class contains the double-modulators, a typical example of discretetime electronic systems with discontinuity and input. First we develop ageneral theory and then we apply it to some examples in order to obtainsets of inadmissible sequences.     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

Oscillation of Higher-Order Delay Differential Equation with Applications to Hyperbolic Equations

In this paper, we consider an odd-order delay differential equation with positive and negative coefficients. New sufficient conditions that guarantee the oscillation of all solutions are presented. Our results extend and improve some known results. Next, these results are used for establishing oscillation criteria for hyperbolic delay differential equations with positive and negative coefficients corresponding to three sets of boundary conditions.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

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