The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavi...The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavior of the solution for the original boundary value problems are discussed. The uniformly effective asymptotic expansion and estimation of solution u(x, ε) were obtained.展开更多
In this paper we study the Goursat problem for semilinear hyperbolicequations with zero boundary condition where the boundary is the characteristic conefor hyperbolic operator. Our result shows that the solution is Li...In this paper we study the Goursat problem for semilinear hyperbolicequations with zero boundary condition where the boundary is the characteristic conefor hyperbolic operator. Our result shows that the solution is Lipschitz and is smoothaway from the characteristic cone.展开更多
This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesia...This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.展开更多
This paper is concerned with Neumann problem for semilinear elliptic equations involving Sobolev critical exponents with limit nonlinearity in boundary condition. By critical point theory and dual variational principl...This paper is concerned with Neumann problem for semilinear elliptic equations involving Sobolev critical exponents with limit nonlinearity in boundary condition. By critical point theory and dual variational principle, the author obtains the existence and multiplicity results.展开更多
The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the sp...The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.展开更多
In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condi...In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.展开更多
A Nitsche-based element-free Galerkin(EFG)method for solving semilinear elliptic problems is developed and analyzed in this paper.The existence and uniqueness of the weak solution for semilinear elliptic problems are ...A Nitsche-based element-free Galerkin(EFG)method for solving semilinear elliptic problems is developed and analyzed in this paper.The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function.A simple iterative scheme is used to deal with the nonlinear integral term.We proved the existence,uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme.A commonly used assumption for approximate space,sometimes called inverse assumption,is proved.Optimal order error estimates in L 2 and H1 norms are proved for the linear and semilinear elliptic problems.In the actual numerical calculation,the characteristic distance h does not appear explicitly in the parameterβintroduced by the Nitsche method.The theoretical results are confirmed numerically。展开更多
This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes...This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes the unit outward normal to boundary aΩ. By vaxiational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved.展开更多
We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy...We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy （which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball） and on Strichartz＇s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.展开更多
The Cauchy problem for the semilinear wave equation has been studied and results show that the problem is locally well-posed in Hs ( Rn ) for s > max [ 0, n/2 - 1]. We extend the results by Lindblad in R3 to R2 and...The Cauchy problem for the semilinear wave equation has been studied and results show that the problem is locally well-posed in Hs ( Rn ) for s > max [ 0, n/2 - 1]. We extend the results by Lindblad in R3 to R2 and R1. The methods used in this paper are different from those of Lindblad and also the methods are more simple.展开更多
A discontinuous Galerkin(DG)scheme for solving semilinear elliptic problem is developed and analyzed in this paper.The DG finite element discretization is first established,then the corresponding well-posedness is pro...A discontinuous Galerkin(DG)scheme for solving semilinear elliptic problem is developed and analyzed in this paper.The DG finite element discretization is first established,then the corresponding well-posedness is provided by using Brouwer’s fixed point method.Some optimal priori error estimates under both DG norm and L^(2)norm are presented,respectively.Numerical results are given to illustrate the efficiency of the proposed approach.展开更多
文摘The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavior of the solution for the original boundary value problems are discussed. The uniformly effective asymptotic expansion and estimation of solution u(x, ε) were obtained.
基金This work is partially supported by National Natural Science Foundation of China.
文摘In this paper we study the Goursat problem for semilinear hyperbolicequations with zero boundary condition where the boundary is the characteristic conefor hyperbolic operator. Our result shows that the solution is Lipschitz and is smoothaway from the characteristic cone.
基金This research was supported by the NASA Nebraska Space Grant(Federal Grant/Award Number 80NSSC20M0112).
文摘This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.
文摘This paper is concerned with Neumann problem for semilinear elliptic equations involving Sobolev critical exponents with limit nonlinearity in boundary condition. By critical point theory and dual variational principle, the author obtains the existence and multiplicity results.
文摘The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.
文摘In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.
基金supported by the Innovation Research Group Project in Universities of Chongqing of China(No.CXQT19018)the National Natural Science Foundation of China(Grant No.11971085)+1 种基金he Natural Science Foundation of Chongqing(Grant Nos.cstc2021jcyj-jqX0011 and cstc2020jcyj-msxm0777)an open project of Key Laboratory for Optimization and Control Ministry of Education,Chongqing Normal University(Grant No.CSSXKFKTM202006)。
文摘A Nitsche-based element-free Galerkin(EFG)method for solving semilinear elliptic problems is developed and analyzed in this paper.The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function.A simple iterative scheme is used to deal with the nonlinear integral term.We proved the existence,uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme.A commonly used assumption for approximate space,sometimes called inverse assumption,is proved.Optimal order error estimates in L 2 and H1 norms are proved for the linear and semilinear elliptic problems.In the actual numerical calculation,the characteristic distance h does not appear explicitly in the parameterβintroduced by the Nitsche method.The theoretical results are confirmed numerically。
文摘This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes the unit outward normal to boundary aΩ. By vaxiational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved.
文摘We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy （which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball） and on Strichartz＇s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.
文摘The Cauchy problem for the semilinear wave equation has been studied and results show that the problem is locally well-posed in Hs ( Rn ) for s > max [ 0, n/2 - 1]. We extend the results by Lindblad in R3 to R2 and R1. The methods used in this paper are different from those of Lindblad and also the methods are more simple.
基金The second and third authors are supported by the National Natural Science Foundation of China(No.12071160)the Guangdong Basic and Applied Basic Research Foundation(No.2019A1515010724)+2 种基金The second author is also supported by the National Natural Science Foundation of China(No.11671159)The third author is also supported by National Natural Science Foundation of China(No.12101250)the Science and Technology Projects in Guangzhou(No.202201010644).
文摘A discontinuous Galerkin(DG)scheme for solving semilinear elliptic problem is developed and analyzed in this paper.The DG finite element discretization is first established,then the corresponding well-posedness is provided by using Brouwer’s fixed point method.Some optimal priori error estimates under both DG norm and L^(2)norm are presented,respectively.Numerical results are given to illustrate the efficiency of the proposed approach.
