To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method...To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method to deal with the ill posedness of the control problem. The determination of the value of the solution of the partial differential equation on the interface——the key of the domain decomposition algorithms——was transformed into a boundary control problem and the ill posedness of the control problem was overcome by regularization. The convergence of the regularizing control solution was proven and the equations which characterize the optimal control were given therefore the value of the unknown solution on the interface of the domain would be obtained by solving a series of coupling equations. Using the boundary control method the domain decomposion algorithm can be carried out.展开更多
This paper presents a comparison among Adomian decomposition method (ADM), Wavelet-Galerkin method (WGM), the fully explicit (1,7) finite difference technique (FTCS), the fully implicit (7,1) finite difference method ...This paper presents a comparison among Adomian decomposition method (ADM), Wavelet-Galerkin method (WGM), the fully explicit (1,7) finite difference technique (FTCS), the fully implicit (7,1) finite difference method (BTCS), (7,7) Crank-Nicholson type finite difference formula (C-N), the fully explicit method (1,13) and 9-point finite difference method, for solving parabolic differential equations with arbitrary boundary conditions and based on weak form functionals in finite domains. The problem is solved rapidly, easily and elegantly by ADM. The numerical results on a 2D transient heat conducting problem and 3D diffusion problem are used to validate the proposed ADM as an effective numerical method for solving finite domain parabolic equations. The numerical results showed that our present method is less time consuming and is easier to use than other methods. In addition, we prove the convergence of this method when it is applied to the nonlinear parabolic equation.展开更多
By the subsuper solutions method, the explosive supersolutions and explosive subsol utions are obtained and the exsistence of explosive solutions is proved on a bounded domain for a class of nonlinear elliptic problem...By the subsuper solutions method, the explosive supersolutions and explosive subsol utions are obtained and the exsistence of explosive solutions is proved on a bounded domain for a class of nonlinear elliptic problems.Then, the exsitence of an entire large solution is proved by the perturbed method.展开更多
This paper is devoted to the investigation of the asymptotic behavior for a class of nonlinear parabolic partial functional differential equations. The boundedness and stability of the solutions are established by the...This paper is devoted to the investigation of the asymptotic behavior for a class of nonlinear parabolic partial functional differential equations. The boundedness and stability of the solutions are established by the upper-lower solution method. Some conditions are obtained by using the semigroup theory, the properties of nonnegative matrices and the techniques of inequalities to determine the asymptotically stable region of the equilibrium.展开更多
In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 ...In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 is a smooth bounded domain in ]1N （N 〉 3）, A, cr 〉 0 are parameters, 0 ≤ μ 〈 μa a 2 ＇ h（x）, KI（X） and K2（x） are positive continuous functions in , 1 〈 q 〈 2, a, β 〉 1 and a ＋ β = 2＊（a,b） （2＊ （a, b） 2N = N-2（1＋a-b） is critical Sobolev-Hardy exponent）. We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters （）, r） belongs to a certain subset of N2.展开更多
The existence and uniqueness of a strong periodic solution of the evolution system describing geophysical flow in bounded domains of RN (N = 3, 4) are proven if external forces are periodic in time and sufficiently sm...The existence and uniqueness of a strong periodic solution of the evolution system describing geophysical flow in bounded domains of RN (N = 3, 4) are proven if external forces are periodic in time and sufficiently small.展开更多
In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u>0 in Ω,N>p>2,where Ω is a domain in RN,possib...In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u>0 in Ω,N>p>2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛV<min ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε0>0 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.展开更多
This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the rele...This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.展开更多
In this paper, we introduce a domain decomposition method with non-matching grids for solving Dirichlet exterior boundary problems by coupling of finite element method （FEM） and natural boundary element method（BEM...In this paper, we introduce a domain decomposition method with non-matching grids for solving Dirichlet exterior boundary problems by coupling of finite element method （FEM） and natural boundary element method（BEM）. We first derive the optimal energy error estimate of the nonconforming approximation generated by this method. Then we apply a Dirichlet-Neumann（D-N） alternating algorithm to solve the coupled discrete system. It will be shown that such iterative method possesses the optimal convergence. The numerical experiments testify our theoretical results.展开更多
This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It...This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It is demonstrated that the answer to this question is no, by constructing a pair of disjoint isospectral planar domains whose boundaries have the same interior Bouligand-Minkowski dimension but are not isometric. In the second part of this paper the authors give the exact two-term asymptotics for the Dirichlet counting functions associated with the examples given here and obtain sharp two sided estimates for the second term of the counting functions. The first result in the second part of the paper shows that the coefficient of the second term is an oscillatory function of λ, which implies that the Weyl-Berry conjecture, for the examples given here, is false. The second result implies that the weaker form of the Weyl-Berry conjecture, for these examples, is true. This in turn means that the interior Bouligand-Minkowski dimension of the examples is a spectral invariant.展开更多
This paper studies an initial-boundary-value problem (IBVP) of the Korteweg-de Vriesequation posed on a finite interval with general nonhomogeneous boundary conditions.Using thestrong Kato smoothing property of the as...This paper studies an initial-boundary-value problem (IBVP) of the Korteweg-de Vriesequation posed on a finite interval with general nonhomogeneous boundary conditions.Using thestrong Kato smoothing property of the associated linear problem,the IBVP is shown to be locallywell-posed in the space H^s(0,1) for any s≥0 via the contraction mapping principle.展开更多
文摘To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method to deal with the ill posedness of the control problem. The determination of the value of the solution of the partial differential equation on the interface——the key of the domain decomposition algorithms——was transformed into a boundary control problem and the ill posedness of the control problem was overcome by regularization. The convergence of the regularizing control solution was proven and the equations which characterize the optimal control were given therefore the value of the unknown solution on the interface of the domain would be obtained by solving a series of coupling equations. Using the boundary control method the domain decomposion algorithm can be carried out.
文摘This paper presents a comparison among Adomian decomposition method (ADM), Wavelet-Galerkin method (WGM), the fully explicit (1,7) finite difference technique (FTCS), the fully implicit (7,1) finite difference method (BTCS), (7,7) Crank-Nicholson type finite difference formula (C-N), the fully explicit method (1,13) and 9-point finite difference method, for solving parabolic differential equations with arbitrary boundary conditions and based on weak form functionals in finite domains. The problem is solved rapidly, easily and elegantly by ADM. The numerical results on a 2D transient heat conducting problem and 3D diffusion problem are used to validate the proposed ADM as an effective numerical method for solving finite domain parabolic equations. The numerical results showed that our present method is less time consuming and is easier to use than other methods. In addition, we prove the convergence of this method when it is applied to the nonlinear parabolic equation.
基金National Natural Science Foundation of China (No.10131050)
文摘By the subsuper solutions method, the explosive supersolutions and explosive subsol utions are obtained and the exsistence of explosive solutions is proved on a bounded domain for a class of nonlinear elliptic problems.Then, the exsitence of an entire large solution is proved by the perturbed method.
基金Supported by NNSFC(19971059)Education Burean of Sichuan Province(01LA43)
文摘This paper is devoted to the investigation of the asymptotic behavior for a class of nonlinear parabolic partial functional differential equations. The boundedness and stability of the solutions are established by the upper-lower solution method. Some conditions are obtained by using the semigroup theory, the properties of nonnegative matrices and the techniques of inequalities to determine the asymptotically stable region of the equilibrium.
文摘In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 is a smooth bounded domain in ]1N （N 〉 3）, A, cr 〉 0 are parameters, 0 ≤ μ 〈 μa a 2 ＇ h（x）, KI（X） and K2（x） are positive continuous functions in , 1 〈 q 〈 2, a, β 〉 1 and a ＋ β = 2＊（a,b） （2＊ （a, b） 2N = N-2（1＋a-b） is critical Sobolev-Hardy exponent）. We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters （）, r） belongs to a certain subset of N2.
基金the Special Funds for Major State Basic Research Projects of China(No.G1999032801) and the National Natural Science Foundation
文摘The existence and uniqueness of a strong periodic solution of the evolution system describing geophysical flow in bounded domains of RN (N = 3, 4) are proven if external forces are periodic in time and sufficiently small.
基金supported by National Natural Science Foundation of China(Grant Nos.11071095 and 11371159)Hubei Key Laboratory of Mathematical Sciences
文摘In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u>0 in Ω,N>p>2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛV<min ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε0>0 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.
基金National Natural Science Foundation of China(No.10071023)MOST and Foundation for University Key TeacherShanghai Priority Academic Discipline
文摘This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.
基金The work of this author was supported by Natural Science Foundation of China(G10371129) The work of this author was supported by the National Basic Research Program of China under the grant G19990328,2005CB321701 the National Natural Science Foundation of China.
文摘In this paper, we introduce a domain decomposition method with non-matching grids for solving Dirichlet exterior boundary problems by coupling of finite element method （FEM） and natural boundary element method（BEM）. We first derive the optimal energy error estimate of the nonconforming approximation generated by this method. Then we apply a Dirichlet-Neumann（D-N） alternating algorithm to solve the coupled discrete system. It will be shown that such iterative method possesses the optimal convergence. The numerical experiments testify our theoretical results.
文摘This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It is demonstrated that the answer to this question is no, by constructing a pair of disjoint isospectral planar domains whose boundaries have the same interior Bouligand-Minkowski dimension but are not isometric. In the second part of this paper the authors give the exact two-term asymptotics for the Dirichlet counting functions associated with the examples given here and obtain sharp two sided estimates for the second term of the counting functions. The first result in the second part of the paper shows that the coefficient of the second term is an oscillatory function of λ, which implies that the Weyl-Berry conjecture, for the examples given here, is false. The second result implies that the weaker form of the Weyl-Berry conjecture, for these examples, is true. This in turn means that the interior Bouligand-Minkowski dimension of the examples is a spectral invariant.
基金supported by the Charles Phelps Taft Memorial Fund of the University of Cincinnatithe Chunhui program (State Education Ministry of China) under Grant No. 2007-1-61006
文摘This paper studies an initial-boundary-value problem (IBVP) of the Korteweg-de Vriesequation posed on a finite interval with general nonhomogeneous boundary conditions.Using thestrong Kato smoothing property of the associated linear problem,the IBVP is shown to be locallywell-posed in the space H^s(0,1) for any s≥0 via the contraction mapping principle.
