By using Karamata regular variation theory and upper and lower solution method,we investigate the existence and the global asymptotic behavior of large solutions to a class of semilinear elliptic equations with nonlin...By using Karamata regular variation theory and upper and lower solution method,we investigate the existence and the global asymptotic behavior of large solutions to a class of semilinear elliptic equations with nonlinear convection terms.In our study,the weight and nonlinearity are controlled by some regularly varying functions or rapid functions,which is very different from the conditions of previous contexts.Our results largely extend the previous works,and prove that the nonlinear convection terms do not affect the global asymptotic behavior of classical solutions when the index of the convection terms change in a certain range.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 deals with a monotone weighted average iterative method for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the ac- celerated monotone iterative method, are construc...This paper deals with a monotone weighted average iterative method for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the ac- celerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone weighted average iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.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
In this paper,we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions.The proof is obtained by a bootstrap argument and maximal r...In this paper,we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions.The proof is obtained by a bootstrap argument and maximal regularity estimates.If n≥10/3m,we also give another proof which does not use maximal regularity estimates.展开更多
In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the M...In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the Morse theory for a C^2-function at both isolated critical point and infinity.展开更多
This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems.Based on elliptic reconstruction approach introduced earlier by Makrid...This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems.Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto[25],a residual based a posteriori error estimators for the state,co-state and control variables are derived.The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements,whereas the piecewise constant functions are employed for the control variable.The temporal discretization is based on the backward Euler method.We derive a posteriori error estimates for the state,co-state and control variables in the L^(∞)(0,T;L^(2)(Ω))-norm.Finally,a numerical experiment is performed to illustrate the performance of the derived estimators.展开更多
In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the orde...In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the order k≤1 RaviartThomas mixed finite element spaces and the control is approximated by piecewise constant element.We derive a posteriori error estimates for the coupled state and control approximations.A numerical example is presented in confirmation of the theory.展开更多
A probabilistic approach is developed to solve semilinear and generalized mixed boundaryvalue problems involving Schrodinger operators. The results obtained in this paper generalize thecorresponding results of [1] and...A probabilistic approach is developed to solve semilinear and generalized mixed boundaryvalue problems involving Schrodinger operators. The results obtained in this paper generalize thecorresponding results of [1] and partly generalize the result of [2] as well.展开更多
In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without th...In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.展开更多
In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouvill...In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.展开更多
In this paper,we consider the following semilinear elliptic equation:■whereΩis an exterior domain in R^N with N≥3,h:Ω×R^+→R is a measurable function,and derive optimal nonexistence results of positive supers...In this paper,we consider the following semilinear elliptic equation:■whereΩis an exterior domain in R^N with N≥3,h:Ω×R^+→R is a measurable function,and derive optimal nonexistence results of positive supersolutions.Our argument is based on a nonexistence result of positive supersolutions of a linear elliptic problem with Hardy potential.We also establish sharp nonexistence results of positive supersolutions to an elliptic system.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
In this paper,we consider the following problem {-Δu（x）＋u（x）=λ（u^p（x）＋h（x））,x∈R^N,u（x）∈h^1（R^N）,u（x）〉0,x∈R^N,（＊）where λ 〉 0 is a parameter,p =（N＋2）/（N—2）.We will prove that there exi...In this paper,we consider the following problem {-Δu（x）＋u（x）=λ（u^p（x）＋h（x））,x∈R^N,u（x）∈h^1（R^N）,u（x）〉0,x∈R^N,（＊）where λ 〉 0 is a parameter,p =（N＋2）/（N—2）.We will prove that there exists a positive constant 0 〈 A＊ 〈 ＋00such that（＊） has a minimal positive solution for λ∈（0,λ＊）,no solution for λ 〉 λ＊,a unique solution for λ = λ＊.Furthermore,（＊） possesses at least two positive solutions when λ∈（0,λ＊） and 3 ≤ N ≤ 5.For N ≥ 6,under some monotonicity conditions of h we show that there exists a constant 0 〈λ＊＊ 〈 λ＊ such that problem（＊）possesses a unique solution for λ∈（0,λ＊＊）.展开更多
This paper contains a generalization of the well–known Palais–Smale andCerami compactness conditions. The compactness condition introduced is used to prove some generalexistence theorems for critical points. Some ap...This paper contains a generalization of the well–known Palais–Smale andCerami compactness conditions. The compactness condition introduced is used to prove some generalexistence theorems for critical points. Some applications are given to differential equations.展开更多
基金Supported by Startup Foundation for Docotors of Weifang University(2016BS04)
文摘By using Karamata regular variation theory and upper and lower solution method,we investigate the existence and the global asymptotic behavior of large solutions to a class of semilinear elliptic equations with nonlinear convection terms.In our study,the weight and nonlinearity are controlled by some regularly varying functions or rapid functions,which is very different from the conditions of previous contexts.Our results largely extend the previous works,and prove that the nonlinear convection terms do not affect the global asymptotic behavior of classical solutions when the index of the convection terms change in a certain range.
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 deals with a monotone weighted average iterative method for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the ac- celerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone weighted average iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.
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In this paper,we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions.The proof is obtained by a bootstrap argument and maximal regularity estimates.If n≥10/3m,we also give another proof which does not use maximal regularity estimates.
文摘In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the Morse theory for a C^2-function at both isolated critical point and infinity.
文摘This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems.Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto[25],a residual based a posteriori error estimators for the state,co-state and control variables are derived.The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements,whereas the piecewise constant functions are employed for the control variable.The temporal discretization is based on the backward Euler method.We derive a posteriori error estimates for the state,co-state and control variables in the L^(∞)(0,T;L^(2)(Ω))-norm.Finally,a numerical experiment is performed to illustrate the performance of the derived estimators.
基金supported by Guangdong Provincial‘Zhujiang Scholar Award Project’National Science Foundation of China 10671163+2 种基金the National Basic Research Program under the Grant 2005CB321703Scientific Research Fund of Hunan Provincial Education Department 06A069Hunan Provincial Innovation Foundation for Postgraduate S2008yjscx04。
文摘In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the order k≤1 RaviartThomas mixed finite element spaces and the control is approximated by piecewise constant element.We derive a posteriori error estimates for the coupled state and control approximations.A numerical example is presented in confirmation of the theory.
基金This project is supported by the National Natural Science Foundation of China
文摘A probabilistic approach is developed to solve semilinear and generalized mixed boundaryvalue problems involving Schrodinger operators. The results obtained in this paper generalize thecorresponding results of [1] and partly generalize the result of [2] as well.
基金supported in part by NSF(Youth) of Shandong Province and NNSF of China
文摘In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.
文摘In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.
基金supported by National Natural Science Foundation of China(Grant Nos.11726614 and 11661045)Jiangxi Provincial Natural Science Foundation(Grant No.20161ACB20007)+4 种基金supported by National Natural Science Foundation of China(Grant Nos.11671175 and 11571200)the Priority Academic Program Development of Jiangsu Higher Education Institutions,Top-notch Academic Programs Project of Jiangsu Higher Education Institutions(Grant No.PPZY2015A013)Qing Lan Project of Jiangsu Provincesupported by National Natural Science Foundation of China(Grant Nos.11726613,11271133 and 11431005)Science and Technology Commission of Shanghai Municipality(STCSM)(Grant No.13d Z2260400)。
文摘In this paper,we consider the following semilinear elliptic equation:■whereΩis an exterior domain in R^N with N≥3,h:Ω×R^+→R is a measurable function,and derive optimal nonexistence results of positive supersolutions.Our argument is based on a nonexistence result of positive supersolutions of a linear elliptic problem with Hardy potential.We also establish sharp nonexistence results of positive supersolutions to an elliptic system.
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.
基金supported by the National Natural Science Foundation of China(No.11201132)Scientific Research Foundation for Ph.D of Hubei University of Technology(No.BSQD12065)supported by the Science Research Project of Hubei Provincial Department of education(No.d200614001)
文摘In this paper,we consider the following problem {-Δu（x）＋u（x）=λ（u^p（x）＋h（x））,x∈R^N,u（x）∈h^1（R^N）,u（x）〉0,x∈R^N,（＊）where λ 〉 0 is a parameter,p =（N＋2）/（N—2）.We will prove that there exists a positive constant 0 〈 A＊ 〈 ＋00such that（＊） has a minimal positive solution for λ∈（0,λ＊）,no solution for λ 〉 λ＊,a unique solution for λ = λ＊.Furthermore,（＊） possesses at least two positive solutions when λ∈（0,λ＊） and 3 ≤ N ≤ 5.For N ≥ 6,under some monotonicity conditions of h we show that there exists a constant 0 〈λ＊＊ 〈 λ＊ such that problem（＊）possesses a unique solution for λ∈（0,λ＊＊）.
文摘This paper contains a generalization of the well–known Palais–Smale andCerami compactness conditions. The compactness condition introduced is used to prove some generalexistence theorems for critical points. Some applications are given to differential equations.
