Let [b,T] be the commutator of the functionb ∈ Lip β (? n ) (0 <β ? 1)and the Calderón-Zygmund singular integral operatorT. The authors study the boundedness properties of [b,T] on the classical Hardy space...Let [b,T] be the commutator of the functionb ∈ Lip β (? n ) (0 <β ? 1)and the Calderón-Zygmund singular integral operatorT. The authors study the boundedness properties of [b,T] on the classical Hardy spaces and the Herz-type Hardy spaces in non-extreme cases. For the boundedness of these commutators in extreme cases, some characterizations are also given. Moreover, the authors prove that these commutators are bounded from Hardy type spaces to the weak Lebesgue or Herz spaces in extreme cases展开更多
This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utiliz...This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.展开更多
A multi-variable non-singular boundary element method (MNBEM) is presented for 2-D potential problems. This method is based on the coincident collocation of non-singular boundary integral equations (BIEs) of the pot...A multi-variable non-singular boundary element method (MNBEM) is presented for 2-D potential problems. This method is based on the coincident collocation of non-singular boundary integral equations (BIEs) of the potential and its derivatives, where the nodal potential derivatives are considered independent of the nodal potential and flux. The system equation is solved to determine the unknown boundary potentials and fluxes, with high accuracy boundary nodal potential derivatives obtained from the solution at the same time. A modified Gaussian elimination algorithm was developed to improve the solution efficiency of the final system equation. Numerical examples verify the validity of the proposed algorithm.展开更多
In this paper,a semilinear pseudo-parabolic equation with a general nonlin-earity and singular potential is considered.We prove the local existence of solution by Galerkin method and contraction mapping theorem.Moreov...In this paper,a semilinear pseudo-parabolic equation with a general nonlin-earity and singular potential is considered.We prove the local existence of solution by Galerkin method and contraction mapping theorem.Moreover,we prove the blow-up of solution and estimate the upper bound of the blow-up time for J(u0)≤0.Finally,we prove the finite time blow-up and estimate the upper bound of blow-up time for J(u0)>0.展开更多
We study radial symmetric point defects with degree k/2 in the 2-D disk or R^(2) in the Q-tensor framework with a singular bulk energy,which is defined by Bingham closure.First,we obtain the existence of solutions for...We study radial symmetric point defects with degree k/2 in the 2-D disk or R^(2) in the Q-tensor framework with a singular bulk energy,which is defined by Bingham closure.First,we obtain the existence of solutions for the profiles of radial symmetric point defects with degree k/2 in the 2-D disk or R^(2).Then,we prove that the solution is stable for |k| = 1 and unstable for |k| > 1.Some identities are derived and utilized throughout the proof of existence and stability/instability.展开更多
Lyapunov＇s first method,extended by Kozlov to nonlinear mechanical systems,is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative fo...Lyapunov＇s first method,extended by Kozlov to nonlinear mechanical systems,is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces.The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position.This fact renders the impossible application of Lyapunov＇s approach in the analysis of the stability because,in the equilibrium position,the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled.It is shown that Kozlov＇s generalization of Lyapunov＇s first method can also be applied in the mentioned cases on the conditions that,besides the known algebraic expression,more are fulfilled.Three theorems on the instability of the equilibrium position are formulated.The results are illustrated by an example.展开更多
In this paper,a new formulation is proposed to evaluate the origin intensity factors(OIFs)in the singular boundary method(SBM)for solving 3D potential problems with Dirichlet boundary condition.The SBM is a strong-for...In this paper,a new formulation is proposed to evaluate the origin intensity factors(OIFs)in the singular boundary method(SBM)for solving 3D potential problems with Dirichlet boundary condition.The SBM is a strong-form boundary discretization collocation technique and is mathematically simple,easy-to-program,and free of mesh.The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions.Traditionally,the inverse interpolation technique(IIT)is adopted to calculate the OIFs on Dirichlet boundary,which is time consuming for large-scale simulation.In recent years,the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary,but the Dirichlet problem remains an open issue.This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary.Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique,the method of fundamental solutions,and the boundary element method.展开更多
In this paper,we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term.By using the logarithmic Sobolev inequality and Hardy's inequality,the ...In this paper,we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term.By using the logarithmic Sobolev inequality and Hardy's inequality,the existence and regularity of multiple nontrivial solutions have been proved.展开更多
In this paper, we establish the existence of at least five distinct solutions to a p-Laplacian problems involving critical exponents and singular cylindrical potential, by using the Nehari manifold, concentration-comp...In this paper, we establish the existence of at least five distinct solutions to a p-Laplacian problems involving critical exponents and singular cylindrical potential, by using the Nehari manifold, concentration-compactness principle and mountain pass theorem展开更多
基金This work was supported by the National 973 Project of China (Grant No.G19990751) the National Natural Science Foundation of China (Grant No. 19131080) the State Education Department Foundation of China (Grant No. 20010027002).
文摘Let [b,T] be the commutator of the functionb ∈ Lip β (? n ) (0 <β ? 1)and the Calderón-Zygmund singular integral operatorT. The authors study the boundedness properties of [b,T] on the classical Hardy spaces and the Herz-type Hardy spaces in non-extreme cases. For the boundedness of these commutators in extreme cases, some characterizations are also given. Moreover, the authors prove that these commutators are bounded from Hardy type spaces to the weak Lebesgue or Herz spaces in extreme cases
文摘This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.
基金Supported by the National Natural Science Foundation of China(No. 10102019) the Special Fund for Returning Scholars in the Chinese Academy of Sciences (No. 20010826214905) and the Ministry of Education of China
文摘A multi-variable non-singular boundary element method (MNBEM) is presented for 2-D potential problems. This method is based on the coincident collocation of non-singular boundary integral equations (BIEs) of the potential and its derivatives, where the nodal potential derivatives are considered independent of the nodal potential and flux. The system equation is solved to determine the unknown boundary potentials and fluxes, with high accuracy boundary nodal potential derivatives obtained from the solution at the same time. A modified Gaussian elimination algorithm was developed to improve the solution efficiency of the final system equation. Numerical examples verify the validity of the proposed algorithm.
基金Supported by National Natural Science Foundation of China(Grant No.11271141).
文摘In this paper,a semilinear pseudo-parabolic equation with a general nonlin-earity and singular potential is considered.We prove the local existence of solution by Galerkin method and contraction mapping theorem.Moreover,we prove the blow-up of solution and estimate the upper bound of the blow-up time for J(u0)≤0.Finally,we prove the finite time blow-up and estimate the upper bound of blow-up time for J(u0)>0.
基金supported by the Basque Government through the BERC PRO-GRAMME 2022-2025 and by the Spanish State Research Agency through Basque Center for Applied Mathematics Severo Ochoa excellence accreditation SEV-2017-0718 and through Project PID2020-114189RB-I00 funded by Agencia Estatal de Investigacion(Grant No.PID2020-114189RB-I00/AEI/10.13039/501100011033)supported by National Natural Science Foundation of China(Grant Nos.11931010 and 12271476)。
文摘We study radial symmetric point defects with degree k/2 in the 2-D disk or R^(2) in the Q-tensor framework with a singular bulk energy,which is defined by Bingham closure.First,we obtain the existence of solutions for the profiles of radial symmetric point defects with degree k/2 in the 2-D disk or R^(2).Then,we prove that the solution is stable for |k| = 1 and unstable for |k| > 1.Some identities are derived and utilized throughout the proof of existence and stability/instability.
基金supported by the Ministry of Science and Technological Development of the Republic of Serbia (Nos. ON174004,ON174016,and TR335006)
文摘Lyapunov＇s first method,extended by Kozlov to nonlinear mechanical systems,is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces.The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position.This fact renders the impossible application of Lyapunov＇s approach in the analysis of the stability because,in the equilibrium position,the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled.It is shown that Kozlov＇s generalization of Lyapunov＇s first method can also be applied in the mentioned cases on the conditions that,besides the known algebraic expression,more are fulfilled.Three theorems on the instability of the equilibrium position are formulated.The results are illustrated by an example.
基金The work described in this paper was supported by the National Science Funds for Distinguished Young Scholars of China(No.11125208)NSFC Funds(Nos.11302069,11372097,11602114 and 11662003)the 111 project under Grant No.B12032.
文摘In this paper,a new formulation is proposed to evaluate the origin intensity factors(OIFs)in the singular boundary method(SBM)for solving 3D potential problems with Dirichlet boundary condition.The SBM is a strong-form boundary discretization collocation technique and is mathematically simple,easy-to-program,and free of mesh.The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions.Traditionally,the inverse interpolation technique(IIT)is adopted to calculate the OIFs on Dirichlet boundary,which is time consuming for large-scale simulation.In recent years,the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary,but the Dirichlet problem remains an open issue.This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary.Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique,the method of fundamental solutions,and the boundary element method.
基金supported by National Natural Science Foundation of China (Grant No.11131005)PHD Programs Foundation of Ministry of Education of China (Grant No. 20090141110003)the Fundamental Research Funds for the Central Universities (Grant No. 2012201020202)
文摘In this paper,we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term.By using the logarithmic Sobolev inequality and Hardy's inequality,the existence and regularity of multiple nontrivial solutions have been proved.
文摘In this paper, we establish the existence of at least five distinct solutions to a p-Laplacian problems involving critical exponents and singular cylindrical potential, by using the Nehari manifold, concentration-compactness principle and mountain pass theorem
