Stability criteria for linear second order delay differential system

Consider second order delay differential system where r is a positive constant and all coefficients are real constants.Our main results are as follows:(1) The maximal length of the delay for which the stability of system (*) is maintained is given in the case where the zero solution of system (*) is asymptotically stable in the absence of delay.(2) The necessary and sufficient criteria for judging that asymptetical stability of system (*) is preserved for an arbitrary large delay are obtained.     

PERIODIC AND ALMOST PERIODIC SOLUTIONS FOR A NON-AUTONOMOUS RESPIRATORY DISEASE MODEL WITH A LAG EFFECT

This paper studies a kind of non-autonomous respiratory disease model with a lag effect.First of all,the permanence and extinction of the system are discussed by using the comparison principle and some differential inequality techniques.Second,it assumes that all coefficients of the system are periodic.The existence of positive periodic solutions of the system is proven,based on the continuation theorem in coincidence with the degree theory of Mawhin and Gaines.In the meantime,the global attract...     

PARAMETER IDENTIFICATION FOR THE DOUBLE POROSITY SYSTEMS IN FRACTURE RESERVOIRS

The inverse parameter problem of the permeable law of the naturally fractured reservoirs with double perosity is of great importance in the exploitation of oil The coefficients in the fluid motion equations are unknown geological parameters. In this paper, the history of matching problem of reservoir systems with double porosity has been discussed from the point of view of the parameter identification of distributed parameter systems, and the problems of identifiability and observability for this system are also dealt with. We have obtained the formulas of Fréchet derivative of the index functional in the identification parameters. These formulas are necessary for calculating identification parameters by a gradient or conjugate gradient method. The conditions of identifiability and observability for the reservoir system with double porosity are given.     

Ground states of a nonlinear Schrdinger system with nonconstant potentials

In the case where either the potentials Vj,μj andβare periodic or Vj are well-shaped andμj andβare anti-well-shaped,existence of a positive ground state of the Schrdinger system -Δu1+V1(x)u1=μ1(x)u31+β(x)u1u22in RN,-Δu2+V2(x)u2=β(x)u21u2+μ2(x)u32in RN,uj∈H1(RN),j=1,2,where N=1,2,3,is proved provided thatβis either small or large in terms of Vj andμj.The system with constant coefficients has been studied extensively in the last ten years,and the nonconstant coefficients case has seldom been studied.It turns out that new technical machineries in the setting of variational methods are needed in dealing with the nonconstant coefficients case.     

Multivalued Stochastic Differential Equations with Non-Lipschitz Coefficients

The existence and uniqueness of solutions to the multivalued stochastic differential equations with non-Lipschitz coefficients are proved, and bicontinuous modifications of the solutions are obtained.     

OSCILLATION AND EXISTENCE OF POSITIVESOLUTIONS OF DIFFERENTIAL EQUATIONSWITH POSITIVE, NEGATIVE COEFFICIENTS

This paper is concerned with the existence of eventually positive solutions and oscillatory solutions of first-order neutral differential equation with positive and negative coefficients. The "sharp" conditions of oscillation for the equation have been obtained, i.e., the conditions are sufficient and necessary when all of the coefficients are constants. These conditions are distinguished from the previous criteria in the literatures, and also weaker than those existing results.     

PERIODIC TRAVELLING-WAVE SOLUTION OF BRUSSELATOR

The Hopf bifurcation Theorem requires that one at first knows all eigenvalues of the linearized system;and a diffieulty in applying the Hopf Theorem is that in the general case we are unable to solve an algebraic equation of degree n (n≥3) with parameter coefficients.This paper firstly proves Theorem 1 which enables us to know the existence of the Hopf bifuecation by direct use of the coefficients of the characteristic equ ation of the linearized system for the case where the characteristic equation of the linearized system has degree 2,3 or 4.Then by using Theorem 1,we gave out the condition for existing a periodic traveling-wave solution of Brusselator in diffusion by Hopf bifurcat on.Finally,we discuss the problem of bifurcation direction and the problem of stability of the bifurcation.     

Strong Solutions for Nonhomogeneous Incompressible Viscous Heat-Conductive Fluids with Non-Newtonian Potential

We consider the Navier-Stokes system with non-Newtonian potential for heat-conducting incompressible fluids in a domain Ω R3. The viscosity, heat conduc- tion coefficients and specific heat at constant volume are allowed to depend smoothly on the density and temperature. We prove the existence of unique local strong solu- tions for all initial data satisfying a natural compatibility condition. The difficult of this type model is mainly that the equations are coupled with elliptic, parabolic and hyper- bolic, and the vacuum of density cause also much trouble, that is, the initial density need not be positive and may vanish in an open set.     

Asymptotic and Oscillatory Behavior of Solutions of First Order Neutral Differential Equations

This paper has made researches on first order neutral differential equations with variable coefficients and several deviations. The asymptotic behavior of nonoscillatory solutions of the equations are discussed. Necessary and sufficient conditions and several sufficient conditions for the oscillations of the equations are obtained. The relevent results in [1—3] are improved and generalized .     

Global wellposedness of magnetohydrodynamics system with temperature-dependent viscosity

The initial boundary value problem of the one-dimensional magneto-hydrodynamics system, when the viscosity, thermal conductivity, and magnetic diffusion coefficients are general smooth functions of temperature, is considered in this article. A unique global classical solution is shown to exist uniquely and converge to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1.     

A Nonlinear Diffusion System with Coupled Nonlinear Boundary Flux

This paper studies a nonlinear diffusion system with coupled nonlinear boundary flux and two kinds of inner sources （positive for the first and negative for the second）, where the four nonlinear mechanisms are described by eight nonlinear parameters. The critical exponent of the system is determined by a complete classification of the eight nonlinear parameters, which is represented via the characteristic algebraic system introduced to the problem.     

BIFURCATION OF BOUNDED SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER

Using the notion of an isolated invariant set and an isolating block, an existence criterion of bifurcation points of nonstationary bounded solutions to ordinary differential systems depending on a parameter is given.     

UNSTABLE SOLUTIONS TO A CLASS OF VECTOR DIFFERENTIAL EQUATIONS OF SIXTH ORDER

In this paper, we consider a class of nonlinear vector differential equations of sixth order. By constructing appropriate Lyapunov functions, the non-existence of periodic solutions is established. Moreover, we provide an example to show the feasibility of our results. Our results extend and improve two related results in the previous literature from scalar cases to vectorial cases.     

A Note to the Cauchy Problem for the Degenerate Parabolic Equations with Strongly Nonlinear Sources

In this note we study the nonexistence of nontrivial global solutions on S = R^N × （0,∞） for the following inequalities：｜u｜t≥△（｜u｜^m-1u）＋｜u｜^q and ｜u｜t≥div（｜△u｜^p-2△｜u｜）＋｜u｜^q.When m,p,q satisfy some given conditions, the nonexistence of nontrivial global solution is proved, without taking their traces on the hyperplans t = 0 into account.     

EXISTENCE OF POSITIVE SOLUTIONS TO A THREE-POINT BOUNDARY VALUE PROBLEM FOR SECOND ORDER DYNAMIC EQUATIONS WITH DERIVATIVE ON TIME SCALES

In this paper, we consider the existence of positive solutions to a three-point boundary value problem for second order dynamic equations with derivative on time scales.Applying Leggett-Williams fixed point theorem, we obtain at least three positive solutions to the problem. An example is also presented to illustrate the applications of the results obtained.     

THREE POSITIVE SOLUTIONS TO BVP FOR p-LAPLACIAN IMPULSIVE FUNCTIONAL DYNAMIC EQUATIONS ON A TIME SCALE

In this paper, we axe interested in the existence of three positive solutions to a BVP for p-Laplacian impulsive functional dynamic equations on a time scale. Using the five-functional fixed theory, we establish a Banach space and an appropriate operator. In this paper, we combine the delta-nabla p-Laplacian BVP with impulsive functional dynamic equations, and obtain some new sufficient conditions for the existence of three positive solutions to the BVP, and our result here generalizes the previous related results.     

SOLUTIONS TO DISCRETE MULTIPARAMETER PERIODIC BOUNDARY VALUE PROBLEMS INVOLVING THE p-LAPLACIAN VIA CRITICAL POINT THEORY

In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [G. Bonanno and P. Candito, Appl.Anal., 88(4)(2009), pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.     

Existence and Uniqueness of BV Solutions for a Class of Degenerate Boltzmann Equations with Measures as Initial Conditions

The existence and uniqueness of the solutions for the Boltzmann equations with measures as initial value are still an open problem which is posed by P. L. Lions （2000）. The aim of this paper is to discuss the Cauchy problem of the system of discrete Boltzmann equations of the form
δtfi＋（fi^mi）=Qi（f1,f2...,fn）,（mi〉1,i-1,...n）
with non-negative finite Radon measures as initial conditions. In particular, the existence and uniqueness of BV solutions for the above problem are obtained.     

SIGN-CHANGING SOLUTIONS FOR SCHRODINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation
-△u ＋ λV（x）u = K（x）｜u｜^p-2u x∈R^N, u→0 as ｜x｜→ ＋∞,
2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V（x） and K（x） satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].     

SOLUTIONS OF A SYSTEM OF FORCED BURGERS EQUATION IN TERMS OF GENERALIZED LAGUERRE POLYNOMIALS

In this article, we obtain explicit solutions of a linear PDE subject to a class of radial square integrable functions with a monotonically increasing weight function |x|n-1eβ|x|2/2,β≥ 0, x ∈ Rn. This linear PDE is obtained from a system of forced Burgers equation via the Cole-Hopf transformation. For any spatial dimension n 1, the solution is expressed in terms of a family of weighted generalized Laguerre polynomials. We also discuss the large time behaviour of the solution of the system of forced Burgers equation.