A KIND OF NONLOCAL PROBLEMS FOR SINGULARLY PERTURBED REACTION DIFFUSION SYSTEMS

The problems of the nonlocal boundary conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems are studied.

Nonlocal controllability for semilinear problems in Banach spaces

If A： D（A） X→X is a densely defined and closed linear operator, which generates a linear semigroup S （t） in Banach space X. The nonlocal control/ability for the following nonlocal semilinear problems： u＇ （t） = Au （t） ＋ Bx（ t） ＋ f（ t, u（t） ）, 0≤t ≤ T with nonlocal initial condition u（0） = u0 ＋ g（u） is discussed in Banach space X. The results show that if semigroup S（t） is strongly continuous, the functionsf and g are compact and the control B is bounded, then it is nonlocally controllable. The nonlocal controllability for the above nonlocal problem is also studied when B and W are unbounded and the semigroup S（t） is compact or strongly continuous. For illustration, a partial differential equation is worked out.

A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels

In this paper,we propose a hybrid spectral method for a type of nonlocal problems,nonlinear Volterra integral equations(VIEs)of the second kind.The main idea is to use the shifted generalized Log orthogonal functions(GLOFs)as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals.This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions.The well-posedness and the related error estimates will be provided.Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.

THE SINGULARLY PERTURBED NONLOCAL BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER NONLINEAR ELLIPTIC EQUATIONS

In this paper,a class of singular perturbation of nonlocal boundary value problems for elliptic partial differential equations of higher order is considered by using the differential inequalities.The uniformly valid asymptotic expansion of solution is obtained.

Exponential Convergence of Finite-dimensional Approximations to Linear Bond-based Peridynamic Boundary Value Problems

In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term, therefore greatly improve the convergence rate derived in literature.

ON SOME MODEL DIFFUSION PROBLEMS WITH A NONLOCAL LOWER ORDER TERM

The authors consider a class of nonlinear parabolic problems where the lower order term isdepending on a weighted integral of the solution, and address the issues of existence, uniqueness,stationary solutions and in some cases asymptotic behaviour.

The Nonlocal Singularly Perturbed Problems forNonlinear Hyperbolic Differential Equation

A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value problems.

Theorems about the Existence of Solutions to Problems with Nonlocal Initial Value

Recently much work has been devoted to nonlocal problems. However, very little has been accomplished in the literature for nonlocal initial problems with nonlinear boundary conditions. It is the purpose of this paper to prove the existence results for solutions to a semilinear parabolic PDE with linear homogeneous boundary conditions, and to other ones with nonlinear boundary conditions, provided the ordered upper and lower solutions are given. Semigroup, fractional order function spaces and generalized Poincare operators play an important role in proving the existence of solutions.

STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets.For the three-dimensional steady non-isentropic compressible Euler system with frictions,we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections.In this article,we are devoted to proving rigorously that a large class of these transonic shock solutions are stable,under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces.This manifests that frictions have a stabilization effect on transonic shocks in ducts,in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts.Except its implications to applications,because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system,we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.

THE NONLOCAL INITIAL PROBLEMS OF A SEMILINEAR EVOLUTION EQUATION

The purpose of this paper is to investigate the existence of solutions to a nonlocal Cauchy problem for an evolution equation. The methods used here include the semigroup methods in proper spaces and Schauder's theorem. And the results are applied to a system of nonlinear partial differential equations with nonlinear boundary conditions.

A CLASS OF NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR NONLINEAR HYPERBOLIC DIFFERENTIAL EQUATION

The nonlocal singularly perturbed problems for the hyperbolic differential equation are considered. Under suitable conditions, using the fixed point theorem, the asymptotic behavior of solution for the initial boundary value problems is studied.

Three Symmetric Positive Solutions for Second-order Nonlocal Boundary Value Problems

Using the Leggett-Williams fixed point theorem, we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u″（t）＋g（t）f（t,u（t））=0,0〈t〈1,u（0）=u（1）=∫01m（s）u（s）ds. where m ∈ L1[0 1], g ： （0, 1）→ [0, ∞） is continuous, symmetric on （0, 1） and maybe singular at t = 0 and t = 1, f： [0, 1] × [0, ∞） → [0, ∞） is continuous and f（-, x） is symmetric on [0, 1] for all x∈ [0, ∞）.

POSITIVE SOLUTIONS TO SYSTEMS OF SECOND ORDER NONLOCAL BOUNDARY VALUE PROBLEMS WITH FIRST ORDER DERIVATIVES

This paper investigates the existence of positive solutions to systems of second order nonlocal boundary value problems with first order derivatives, in which the nonlinear term is not required to be continuous and involves first order derivatives. The main tool used in this paper is a fixed point index theory in a cone.

Multiplicity of Solutions for a Class of Kirchhoff Type Problems

In this paper we apply the （variant） fountain theorems to study the symmetric nonlinear Kirch- hoff nonlocal problems. Under the Ambrosetti-Rabinowitz＇s 4-superlinearity condition, or no Ambrosetti- Rabinowitz＇s 4-superlinearity condition, we present two results of existence of infinitely many large energy solutions, respectively.

NONLOCAL PROBLEM FOR SINGULARLY PERTURBED REACTION DIFFUSION SYSTEMS

In this paper, the problems of the nonlocal initial conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solutions for the initial boundary value problems are studied.

Blow-Up of Solutions for a Singular Nonlocal Viscoelastic Equation

We study the nonlinear one-dimensional viscoelastic nonlocal problem： uff-1/x（xux）x＋∫t0g（t-s）1/x）xux（x,s））xds=｜u｜p-2u,with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].

NONLOCAL INITIAL PROBLEM FOR NONLINEAR NONAUTONOMOUS DIFFERENTIAL EQUATIONS IN A BANACH SPACE

The nonlocal initial problem for nonlinear nonautonomous evolution equati-ons in a Banach space is considered. It is assumed that the nonlinearities havethe local Lipschitz properties. The existence and uniqueness of mild solutionsare proved. Applications to integro-differential equations are discussed.The main tool in the paper is the normalizing mapping (the generalizednorm).

Positive Ground State Solutions fora Critical Nonlocal Problem in Dimension Three

In this paper,we are interested in the following nonlocal problem with critical exponent{-(a-b∫Ω|▽u|2dx)△u=λ|u|p-2+|u|4u,u=0,x∈Ω,x∂∈Ω,where a,b are positive constants,2<p<6,Ωis a smooth bounded domain in R^(3)andλ>O is a parameter.By variational methods,we prove that problem has a positive ground state solution up forλ>0 sufficiently large.Moreover,we take b as a parameter and study the asymptotic behavior of ub when b↘O.

On a Nonlocal Problem Modelling Ohmic Heating in Planar Domains

In this paper, we consider the nonlocal problem of the formand the associated nonlocal stationary problemwhere λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problem has a unique solution if and only if λ 〈 2｜Ω｜2, and for λ = 2｜Ω｜2, the solution of the nonlocal parabolic problem grows up globally to infinity as t→∞.