A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the...A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.展开更多
We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. ...We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.展开更多
This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of suff...This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.展开更多
We study positive solutions to the fractional semi-linear elliptic equation(−∆)σu=K(x)u n+2σn−2σin B2\{0}with an isolated singularity at the origin,where K is a positive function on B2,the punctured ball B2\{0}⊂Rn ...We study positive solutions to the fractional semi-linear elliptic equation(−∆)σu=K(x)u n+2σn−2σin B2\{0}with an isolated singularity at the origin,where K is a positive function on B2,the punctured ball B2\{0}⊂Rn with n>2,σ∈(0,1),and(−∆)σis the fractional Laplacian.In lower dimensions,we show that for any K∈C1(B2),a positive solution u always satisfies that u(x)6 C|x|−(n−2σ)/2 near the origin.In contrast,we construct positive functions K∈C1(B2)in higher dimensions such that a positive solution u could be arbitrarily large near the origin.In particular,these results also apply to the prescribed boundary mean curvature equations on B n+1.展开更多
The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions ...The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions between two breathers,a breather/lump and line solitons as well as lump molecules for the(2+1)-dimensional elliptic Toda equation.Based on the N-soliton solution,we obtain the hybrid solutions consisting of line solitons,breathers and lumps.Through the asymptotic analysis of these hybrid solutions,we derive the phase shifts of the breather,lump and line solitons before and after the interaction between a breather/lump and line solitons.By making the phase shifts infinite,we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons.Through the asymptotic analysis of these resonant solutions,we demonstrate that the resonant interactions exhibit the fusion,fission,time-localized breather and rogue lump phenomena.Utilizing the velocity resonance method,we obtain lump–soliton,lump–breather,lump–soliton–breather and lump–breather–breather molecules.The above works have not been reported in the(2+1)-dimensional discrete nonlinear wave equations.展开更多
In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the m...In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.展开更多
In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate pa...In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.展开更多
In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogene...In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.展开更多
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
The singularly perturbed generalized boundary value problems far the quasi- linear elliptic equation of higher order are considered. Under suitable conditions, the existence, uniqueness and asymptotic behavior of the ...The singularly perturbed generalized boundary value problems far the quasi- linear elliptic equation of higher order are considered. Under suitable conditions, the existence, uniqueness and asymptotic behavior of the generalized solution for the Dirichlet problems are studied.展开更多
By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic functio...By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.展开更多
By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of ...By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.展开更多
It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to ...It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).展开更多
Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct dou...Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.展开更多
An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdVequation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu...An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdVequation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu's allresults are obtained. When the modulus m → 1 or 0, we can find the corresponding six solitary wave solutions and sixtrigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian ellipticfunction solutions and can be applied to other nonlinear differential equations.展开更多
An existent theorem is obtained for nonzero W-1,W-2(R-N) solutions of the following equations on R-N -Delta u + b(x)u = f(x,u), x is an element of R-N, where b is periodic for some variables and coercive for the other...An existent theorem is obtained for nonzero W-1,W-2(R-N) solutions of the following equations on R-N -Delta u + b(x)u = f(x,u), x is an element of R-N, where b is periodic for some variables and coercive for the others, f is superlinear.展开更多
The asymptotic behavior at infinity and an estimate of positive radial solutions of the equation △u + sum from i=1 to k cirli upi = 0, x ∈ Rn,(0.1)are obtained and the structure of separation property of positive...The asymptotic behavior at infinity and an estimate of positive radial solutions of the equation △u + sum from i=1 to k cirli upi = 0, x ∈ Rn,(0.1)are obtained and the structure of separation property of positive radial solutions of Eq. (0.1) with different initial data α is discussed.展开更多
In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansio...In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.展开更多
(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equ...(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.展开更多
The nonlinear nonlocal singularly perturbed boundary value problems for elliptic equation with boundary perturbation was considered.Under suitable conditions,firstly,the outer solution of the original problem is obtai...The nonlinear nonlocal singularly perturbed boundary value problems for elliptic equation with boundary perturbation was considered.Under suitable conditions,firstly,the outer solution of the original problem is obtained,secondly,using the stretched variable,the composing expansion method and the expanding theory of power series the boundary layer is constructed,finally,using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems is studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.展开更多
文摘A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.
文摘We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.
文摘This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.
文摘We study positive solutions to the fractional semi-linear elliptic equation(−∆)σu=K(x)u n+2σn−2σin B2\{0}with an isolated singularity at the origin,where K is a positive function on B2,the punctured ball B2\{0}⊂Rn with n>2,σ∈(0,1),and(−∆)σis the fractional Laplacian.In lower dimensions,we show that for any K∈C1(B2),a positive solution u always satisfies that u(x)6 C|x|−(n−2σ)/2 near the origin.In contrast,we construct positive functions K∈C1(B2)in higher dimensions such that a positive solution u could be arbitrarily large near the origin.In particular,these results also apply to the prescribed boundary mean curvature equations on B n+1.
基金the National Natural Science Foundation of China(Grant Nos.12061051 and 11965014)。
文摘The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions between two breathers,a breather/lump and line solitons as well as lump molecules for the(2+1)-dimensional elliptic Toda equation.Based on the N-soliton solution,we obtain the hybrid solutions consisting of line solitons,breathers and lumps.Through the asymptotic analysis of these hybrid solutions,we derive the phase shifts of the breather,lump and line solitons before and after the interaction between a breather/lump and line solitons.By making the phase shifts infinite,we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons.Through the asymptotic analysis of these resonant solutions,we demonstrate that the resonant interactions exhibit the fusion,fission,time-localized breather and rogue lump phenomena.Utilizing the velocity resonance method,we obtain lump–soliton,lump–breather,lump–soliton–breather and lump–breather–breather molecules.The above works have not been reported in the(2+1)-dimensional discrete nonlinear wave equations.
文摘In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.
文摘In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.
基金supported by Ministry of Education and Training(Vietnam),under grant number B2023-SPS-01。
文摘In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.
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.
文摘The singularly perturbed generalized boundary value problems far the quasi- linear elliptic equation of higher order are considered. Under suitable conditions, the existence, uniqueness and asymptotic behavior of the generalized solution for the Dirichlet problems are studied.
基金Project supported by the State Key Program for Basic Research of China (Grant No 2004CB418304)the National Natural Science Foundation of China (Grant No 40405010)
文摘By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.
基金Project supported by the National Natural Science Foundation of China(Grant No 10461006), the High Education Science Research Program(Grant No NJ02035) of Inner Mongolia Autonomous Region, Natural Science Foundation of Inner Mongolia Autonomous Region(Grant No 2004080201103) and the Youth Research Program of Inner Mongolia Normal University(Grant No QN005023).
文摘By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.
文摘It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).
文摘Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.
文摘An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdVequation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu's allresults are obtained. When the modulus m → 1 or 0, we can find the corresponding six solitary wave solutions and sixtrigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian ellipticfunction solutions and can be applied to other nonlinear differential equations.
文摘An existent theorem is obtained for nonzero W-1,W-2(R-N) solutions of the following equations on R-N -Delta u + b(x)u = f(x,u), x is an element of R-N, where b is periodic for some variables and coercive for the others, f is superlinear.
基金Supported by the Natural Science Foundation of China(10901126)
文摘The asymptotic behavior at infinity and an estimate of positive radial solutions of the equation △u + sum from i=1 to k cirli upi = 0, x ∈ Rn,(0.1)are obtained and the structure of separation property of positive radial solutions of Eq. (0.1) with different initial data α is discussed.
文摘In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.
文摘(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.
文摘The nonlinear nonlocal singularly perturbed boundary value problems for elliptic equation with boundary perturbation was considered.Under suitable conditions,firstly,the outer solution of the original problem is obtained,secondly,using the stretched variable,the composing expansion method and the expanding theory of power series the boundary layer is constructed,finally,using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems is studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.