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 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.展开更多
The purpose of this paper is to investigate a nonlocal Dirichlet problem with(p(x),q(x))-Laplacian-like operator originated from a capillary phenomena.Using the variational methods and the critical point theory,we est...The purpose of this paper is to investigate a nonlocal Dirichlet problem with(p(x),q(x))-Laplacian-like operator originated from a capillary phenomena.Using the variational methods and the critical point theory,we establish the existence of infinitely many weak solutions for this problem.展开更多
In the paper,a reduced basis(RB)method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed.Because of the lack of sparsity of discretized nonlocal systems c...In the paper,a reduced basis(RB)method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed.Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation(PDE)systems,model reduction for nonlocal systems becomes more critical.The method of snapshots and greedy(MOS-greedy)algorithm of RB method is developed for nonlocal problems with random inputs,which provides an efficient and reliable approximation of the solution.A major challenge lies in the excessive influence of the time domain on the model reduction process.To address this,the Fourier transform is applied to convert the original time-dependent parabolic equation into a frequency-dependent elliptic equation,where variable frequencies are independent.This enables parallel computation for approximating the solution in the frequency domain.Finally,the proposed MOS-greedy algorithm is applied to the nonlocal diffusion problems.Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantly improves computational efficiency.展开更多
A time multipoint nonlocal problem for a Schrödinger equation driven by a cylindrical Q-Wiener process is presented.The initial value depends on a finite number of future values.Existence and uniqueness of a solu...A time multipoint nonlocal problem for a Schrödinger equation driven by a cylindrical Q-Wiener process is presented.The initial value depends on a finite number of future values.Existence and uniqueness of a solution formulated as a mild solution is obtained.A single-step implicit Euler-Maruyama difference scheme,a Rothe-Maryuama scheme,is suggested as a numerical solution.Convergence rate for the solution of the difference scheme is established.The theoretical statements for the solution of this difference scheme is supported by a numerical example.展开更多
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...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.展开更多
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...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.展开更多
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 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.展开更多
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 u...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→∞.展开更多
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 a...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.展开更多
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 ass...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.展开更多
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 probl...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.展开更多
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 ...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.展开更多
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 s...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.展开更多
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 th...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.展开更多
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 boundar...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.展开更多
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 s...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.展开更多
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 ...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).展开更多
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...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, ∞）.展开更多
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 in...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.展开更多
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...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.展开更多
文摘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.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.31920180041)the Fund for Talent Introduction of Northwest Minzu University(Grant No.xbmuyjrc201907)。
文摘The purpose of this paper is to investigate a nonlocal Dirichlet problem with(p(x),q(x))-Laplacian-like operator originated from a capillary phenomena.Using the variational methods and the critical point theory,we establish the existence of infinitely many weak solutions for this problem.
基金supported by the Guangdong Basic and Applied Basic Research Foundation,China(Grant 2024A1515012548)supported by the National Natural Science Foundation of China(Grant 12401567)+1 种基金by the 2023 Guangzhou Basic and Applied Basic Research Project(Grant 2023A04J0035)by the Talent Special Projects of School-level Scientific Research Programs under Guangdong Poiytechnic Normal University(Grant 2022SDKYA025).
文摘In the paper,a reduced basis(RB)method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed.Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation(PDE)systems,model reduction for nonlocal systems becomes more critical.The method of snapshots and greedy(MOS-greedy)algorithm of RB method is developed for nonlocal problems with random inputs,which provides an efficient and reliable approximation of the solution.A major challenge lies in the excessive influence of the time domain on the model reduction process.To address this,the Fourier transform is applied to convert the original time-dependent parabolic equation into a frequency-dependent elliptic equation,where variable frequencies are independent.This enables parallel computation for approximating the solution in the frequency domain.Finally,the proposed MOS-greedy algorithm is applied to the nonlocal diffusion problems.Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantly improves computational efficiency.
文摘A time multipoint nonlocal problem for a Schrödinger equation driven by a cylindrical Q-Wiener process is presented.The initial value depends on a finite number of future values.Existence and uniqueness of a solution formulated as a mild solution is obtained.A single-step implicit Euler-Maruyama difference scheme,a Rothe-Maryuama scheme,is suggested as a numerical solution.Convergence rate for the solution of the difference scheme is established.The theoretical statements for the solution of this difference scheme is supported by a numerical example.
基金the National Natural Science Foundation of China(No.10674024)
文摘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.
基金The project is supported by the Natural Science Foundation of China (No. 10071048).
文摘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.
基金supported by National Natural Science Foundation of China(No.11871152)Natural Science Foundation of Fujian Province(No.2021J01330).
文摘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.
基金Supported by the Foundation for Young Talents in College of Anhui Province （Grant No. 2011SQRL115）, Program sponsored for Scientific Innovation Research of College Graduate in Jangsu Province （Grant No. 181200000649）, the courses building projects of Anhui Science and Technology University （Grant No. ZDKC1121）, the pre-research project of Anhui Science and Technology University （Grant No. ZRC2012308） and Training Fund of Xi＇an University of Science and TechnologyAcknowledgements The authors are grateful to the anonymous referees for their careful reading of the manuscript and numerous suggestions for its improvement.
文摘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→∞.
文摘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.
文摘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.
基金The project is supported by the National Natural Science Foundation of China(10071048)
文摘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.
基金Supported partly by Research Grant 19531060 of China NSFGrant 97024811 of the Foundation of Doctoral Program
文摘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.
基金Project supported by the Swiss National Science Foundation under the contract#20-67618.02.
文摘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.
文摘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.
基金the National Natural Science Foundation of China (No. 10071048>
文摘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.
基金This work was supported in part by National Nature Science Foundation of China(11371141 and 11871218)by Science and Technology Commission of Shanghai Municipality(18dz2271000).
文摘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.
文摘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).
基金Supported by the National Natural Science Foundation of Zhejiang Province of China(No.Y605144)the Science Research Foundation of Educational Department of Zhejiang Province of China(No.200804671)
文摘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, ∞）.
基金supported by Natural Science Foundation of Anhui University of Architecture(20071201)
文摘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.
文摘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.
