This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽...This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.展开更多
In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-(a+bfR^3|↓△u|^2dx)△u+V(x)u=f(x,u),x∈R^3, u∈H^2(R3),where a, b 〉 0 are constants and the primitive...In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-(a+bfR^3|↓△u|^2dx)△u+V(x)u=f(x,u),x∈R^3, u∈H^2(R3),where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.展开更多
By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established re...By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.展开更多
In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three...In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three spatial dimensions.The newly constructed element is proved to be convergent for a model biharmonic equation.展开更多
In this paper, we first present constructing a Lyapunov function for (1. 1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡ 0,and the boundedness result of the sol...In this paper, we first present constructing a Lyapunov function for (1. 1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡ 0,and the boundedness result of the solutions of (1 .1 ) for case p≠0. These results improve sveral well-known results.展开更多
This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays...This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.展开更多
In this paper, we consider the following nonlinear elliptic problem : △2u =|u|n-4^-u+u|u|q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/(n-4)...In this paper, we consider the following nonlinear elliptic problem : △2u =|u|n-4^-u+u|u|q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/(n-4),(12-n)/(n-4)].We study the solutions which concentrate around two points of Ω. We prove that the concentration speeOs are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω=(Ωa)a a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for u〉0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Ω.展开更多
In this paper, we consider the following fourth order ordinary differential equation x(4)(t) = f(t,x(t),x (t),x (t),x (t)), t ∈ (0,1) (E) with the four-point boundary value conditions: x(0) = x(1) = 0, αx (ξ1) - β...In this paper, we consider the following fourth order ordinary differential equation x(4)(t) = f(t,x(t),x (t),x (t),x (t)), t ∈ (0,1) (E) with the four-point boundary value conditions: x(0) = x(1) = 0, αx (ξ1) - βx (ξ1) = 0, γx (ξ2) + δx (ξ2) = 0, (B) where 0 < ξ1 < ξ2 < 1. At the resonance condition αδ + βγ + αγ(ξ2 - ξ1) = 0, an existence result is given by using the coincidence degree theory. We also give an example to demonstrate the result.展开更多
The boundary controllability of the fourth order Schr5dinger equation in a bounded domain is studied. By means of an L2-Neumann boundary control, the authors prove that the solution is exactly controllable in H-2(12...The boundary controllability of the fourth order Schr5dinger equation in a bounded domain is studied. By means of an L2-Neumann boundary control, the authors prove that the solution is exactly controllable in H-2(12) for an arbitrarily small time. The method of proof combines both the HUM (Hilbert Uniqueness Method) and multiplier techniques.展开更多
By the Lyapunov functional approach, some better results on the asymptotic stabiBy the Lyapunov functional approach, some better results on the asymptotic stability and global asymptotic stability of zero solution to ...By the Lyapunov functional approach, some better results on the asymptotic stabiBy the Lyapunov functional approach, some better results on the asymptotic stability and global asymptotic stability of zero solution to a certain fourth-order non-linear differential equation with delay are obtained.展开更多
In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities wi...In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.展开更多
In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of ...In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of eigenfunctions of the Laplacian.展开更多
In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R^N, as well as Liouville type results for some solutions converging to the same value at in...In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R^N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.展开更多
In this paper, we consider a certain fourth-order nonlinear ordinary differential equation. Some sufficient conditions which guarantee the existence of at least one ω-periodic solution to the system are obtain.
In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mas...In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mass conservation, energy decay property, unique solvability and L~ convergence of these two schemes are proved. Numerical results demon- strate the good approximation of the fourth order equation and confirm reliability of these two schemes.展开更多
In this paper, we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly weighted Sobolev spaces, which serve as im...In this paper, we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly weighted Sobolev spaces, which serve as important tools in analysis of numerical quadratures, and numerical methods of differential and integral equations. Then we propose Jacobi pseudospectral schemes for several singular problems and multiple-dimensional problems of fourth order. Numerical results demonstrate the spectral accuracy of these schemes, and coincide well with theoretical analysis.展开更多
In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress ef...In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress effect. An explicit estimate formula in terms of the elastic constants of the anisotropic materials is presented. Finally, a numerical example for an end-loaded, off-axis, graphite-epoxy strip is given to illustrate the results.展开更多
基金supported by the National Natural Science Foundation of China(12271296,12271195).
文摘This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.
基金supported by Natural Science Foundation of China(11271372)Hunan Provincial Natural Science Foundation of China(12JJ2004)
文摘In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-(a+bfR^3|↓△u|^2dx)△u+V(x)u=f(x,u),x∈R^3, u∈H^2(R3),where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.
文摘By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.
文摘In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three spatial dimensions.The newly constructed element is proved to be convergent for a model biharmonic equation.
文摘In this paper, we first present constructing a Lyapunov function for (1. 1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡ 0,and the boundedness result of the solutions of (1 .1 ) for case p≠0. These results improve sveral well-known results.
基金Project supported by the National Natural Science Foundation of China (No.10271034)the Natural Science Foundation of Heitongjiang Province of China (No.A2007-02)
文摘This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.
文摘In this paper, we consider the following nonlinear elliptic problem : △2u =|u|n-4^-u+u|u|q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/(n-4),(12-n)/(n-4)].We study the solutions which concentrate around two points of Ω. We prove that the concentration speeOs are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω=(Ωa)a a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for u〉0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Ω.
基金the Master’s Research Fund of Suzhou University (No.2008yss19)
文摘In this paper, we consider the following fourth order ordinary differential equation x(4)(t) = f(t,x(t),x (t),x (t),x (t)), t ∈ (0,1) (E) with the four-point boundary value conditions: x(0) = x(1) = 0, αx (ξ1) - βx (ξ1) = 0, γx (ξ2) + δx (ξ2) = 0, (B) where 0 < ξ1 < ξ2 < 1. At the resonance condition αδ + βγ + αγ(ξ2 - ξ1) = 0, an existence result is given by using the coincidence degree theory. We also give an example to demonstrate the result.
基金supported by the Fundamental Research Funds for the Central Universities (No. XDJK2009C099)the National Natural Science Foundation of China (Nos. 11001018,11026111)the Specialized Research Fund for the Doctoral Program of Higher Education (No. 201000032006)
文摘The boundary controllability of the fourth order Schr5dinger equation in a bounded domain is studied. By means of an L2-Neumann boundary control, the authors prove that the solution is exactly controllable in H-2(12) for an arbitrarily small time. The method of proof combines both the HUM (Hilbert Uniqueness Method) and multiplier techniques.
基金supported by the National Natural Science Foundation of China(10461006)Basic Subject Foundation of Changzhou University(JS201004)
文摘By the Lyapunov functional approach, some better results on the asymptotic stabiBy the Lyapunov functional approach, some better results on the asymptotic stability and global asymptotic stability of zero solution to a certain fourth-order non-linear differential equation with delay are obtained.
基金supported by National Natural Science Foundation of China(Grant No.11401310)supported by National Natural Science Foundation of China(Grant No.11531005).
文摘In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10671040, 10831007, 10801041)the National Excellent Doctoral Dissertation of China (Grant No. 200522)the New Century Excellent Talents in University (Grant No. 06-0359)
文摘In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of eigenfunctions of the Laplacian.
基金carried out in the framework of the Labex Archimède(ANR-11-LABX-0033)the A*MIDEX project(ANR-11-IDEX-0001-02)+6 种基金funded by the "Investissements d’Avenir" French Government program managed by the French National Research Agency(ANR)funding from the European Research Council under the European Union’s Seventh Framework Programme(FP/2007-2013)ERC Grant Agreement n.321186-ReaDiReaction-Diffusion Equations,Propagation and Modelling and from the ANR NONLOCAL project(ANR-14-CE25-0013)supported by INRIA-Team MEPHYSTOMIS F.4508.14(FNRS)PDR T.1110.14F(FNRS)ARC AUWB-2012-12/17-ULB1-IAPAS
文摘In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R^N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.
文摘In this paper, we consider a certain fourth-order nonlinear ordinary differential equation. Some sufficient conditions which guarantee the existence of at least one ω-periodic solution to the system are obtain.
文摘In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mass conservation, energy decay property, unique solvability and L~ convergence of these two schemes are proved. Numerical results demon- strate the good approximation of the fourth order equation and confirm reliability of these two schemes.
基金The work of these authors is supported in part by NSF of China, N.10471095, Science Foundation of Shanghai N.04JC14062, Special Funds for Doctorial Authorities of Chinese Education Ministry N.20040270002, Shanghai Leading Academic Discipline Project N.T0401, E-institutes of Shanghai Municipal Education Commission, N.E03004, Special Funds for Major Specialities and Fund N.04DB15 of Shanghai Education Commission.
文摘In this paper, we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly weighted Sobolev spaces, which serve as important tools in analysis of numerical quadratures, and numerical methods of differential and integral equations. Then we propose Jacobi pseudospectral schemes for several singular problems and multiple-dimensional problems of fourth order. Numerical results demonstrate the spectral accuracy of these schemes, and coincide well with theoretical analysis.
文摘In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress effect. An explicit estimate formula in terms of the elastic constants of the anisotropic materials is presented. Finally, a numerical example for an end-loaded, off-axis, graphite-epoxy strip is given to illustrate the results.