Soft materials with semi-linear strain energy function can be used as smart transformation media to manipulate elastic waves via finite pre-deformation. However, the intrinsic cons train ts involved in such materials ...Soft materials with semi-linear strain energy function can be used as smart transformation media to manipulate elastic waves via finite pre-deformation. However, the intrinsic cons train ts involved in such materials limit the shapes of t ransformation devices to very sim - pie cases. In this work, combining theoretical and numerical analyses, we report an approach of achieving the in-plane elastodynamic cloak with arbitrary shape. We demonstrate that with the appropriate out-of^plane st retch applied on the semi-linear material, cloaking effec t can be achieved for both P- and SV-waves in the symmetrie plane of a 3D domain, and the performance of the cloak with arbitrary cross section can be guaranteed for relatively small in-plane rot at ion. In addition, we propose an empirical formula to predic t the deformation limit of the cloaks with semi-linear materials. This work may stimulate the experimental research on softmatter- based transformation devices. Potential applications can be anticipated in nondestructive testing, structure impact protection, biomedical imaging and soft robotics.展开更多
An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variab...An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variables: the scalar unknown, its gradient and its flux(coefficient times the gradient), simultaneously. We also prove the existence and uniqueness of semi-discrete solution. Finally, we obtain some numerical results to illustrate the efficiency of the method.展开更多
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
In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-...In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-up result for the semi-linear wave equation when the exponent of p is under certain conditions.Meanwhile,we derive an upper bound of the lifespan of solutions to the Cauchy problem for the semi-linear wave equation.展开更多
A class of semi-linear Robin problem is considered. Under appropriate assumptions, the existence and asymptotic behavior of its solution are studied more carefully. Using stretched variables, the formal asymptotic exp...A class of semi-linear Robin problem is considered. Under appropriate assumptions, the existence and asymptotic behavior of its solution are studied more carefully. Using stretched variables, the formal asymptotic expansion of solution for the problem is constructed and the uniform validity of the solution is obtained by using the method of upper and lower solution.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
In this paper we shall generalize the definition given in [1] for Lipschitz condition and contractions for functions on a non-metrizable space, besides we shall give more properties of semi-linear uniform spaces.
In this paper, by using the topological degree method and some limiting arguments, the existence of admissible periodic bouncing solutions for a class of non-conservative semi-linear impact equations is proved.
The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achie...The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achieving internal and boundary regional controllability. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) and Schauder’s fixed point theorem. We give a numerical example developed in internal and boundary sub region. These numerical illustrations show the efficiency of the approach and lead to conjectures.展开更多
In this paper, we investigate the effect of weight function in the nonlinear part on global solvability of the Cauchy problem for a class of semi-linear hyperbolic equations with damping.
In this paper, by using the techniques of differential inequalities, we prove the existence of the solutions of a singularly perturbed boundary value problem for the third order semilinear differential equation with a...In this paper, by using the techniques of differential inequalities, we prove the existence of the solutions of a singularly perturbed boundary value problem for the third order semilinear differential equation with a turning point.展开更多
This paper addresses the problem of identifying the unknown diffusion coefficient in a semilinear parabolic equation with incomplete initial condition.We propose an optimal control approach using the no-regret control...This paper addresses the problem of identifying the unknown diffusion coefficient in a semilinear parabolic equation with incomplete initial condition.We propose an optimal control approach using the no-regret control method and the adapted low-regret control.Our approach provides a fullcharacterisation of the unknown diffusion coefficient independent of the missing initial condition.We also present an optimality system that describes the adapted low-regret control and use it to find a fuil description of the no-regret control by taking the limit of the sequence of adapted low-regret controls.展开更多
This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stoc...This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.展开更多
In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation ...In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation as t tends to infinity. The long asymptotic behavior of its discrete solution is obtained which is analogous to that of its continuous solution. At last, a few results are also presented for Crank-Nicolson scheme.展开更多
The authors study the Cauchy problem for the semi-linear damped wave equation utt-△u＋b（t）ut=f（u）,u（0,χ）=u0（χ）,ut（0,χ）=u1（χ） in any space dimension n ≥ 1. It is assumed that the time-dependent dampin...The authors study the Cauchy problem for the semi-linear damped wave equation utt-△u＋b（t）ut=f（u）,u（0,χ）=u0（χ）,ut（0,χ）=u1（χ） in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b（t）〉 0 is effective, and in particular tb（t） →∞ as t →∞. The global existence of small energy data solutions for｜f（u）｜≈｜u｜^p in the supercritical case of p 〉 1＋ 2/n and p ≤n/n-2 for n ≥ 3 is proved.展开更多
In this paper,we propose an immersed finite volume element method for solving the semi-linear elliptic interface problems with non-homogeneous jump conditions.Furthermore,two-grid techniques are used to improve the co...In this paper,we propose an immersed finite volume element method for solving the semi-linear elliptic interface problems with non-homogeneous jump conditions.Furthermore,two-grid techniques are used to improve the computational efficiency.In this way,we only need to solve a non-linear system on the coarse grid,and a linear system on the fine grid.Numerical results illustrate that the proposed method can solve the semi-linear elliptic interface problems efficiently.Approximate secondorder accuracy for the solution in the L¥norm can be obtained for the considered examples.展开更多
In this paper, in the sense of the definition of almost periodicity given by H.Bohr using fixed-point principle, we investigate the existence and uniqueness of quadratic mean almost periodic solutions to semi-linear s...In this paper, in the sense of the definition of almost periodicity given by H.Bohr using fixed-point principle, we investigate the existence and uniqueness of quadratic mean almost periodic solutions to semi-linear stochastic integro-differential evolution equations associated with abstract Volterra equations. Some examples are also given to illustrate our theory.展开更多
基金National Natural Science Foundation of China (Grant Nos. 11472044, 11521062, 11602294, 11632003)the Chinese Universities Scientific Fund (Grant No. 2019TC134).
文摘Soft materials with semi-linear strain energy function can be used as smart transformation media to manipulate elastic waves via finite pre-deformation. However, the intrinsic cons train ts involved in such materials limit the shapes of t ransformation devices to very sim - pie cases. In this work, combining theoretical and numerical analyses, we report an approach of achieving the in-plane elastodynamic cloak with arbitrary shape. We demonstrate that with the appropriate out-of^plane st retch applied on the semi-linear material, cloaking effec t can be achieved for both P- and SV-waves in the symmetrie plane of a 3D domain, and the performance of the cloak with arbitrary cross section can be guaranteed for relatively small in-plane rot at ion. In addition, we propose an empirical formula to predic t the deformation limit of the cloaks with semi-linear materials. This work may stimulate the experimental research on softmatter- based transformation devices. Potential applications can be anticipated in nondestructive testing, structure impact protection, biomedical imaging and soft robotics.
基金Supported by the National Natural Science Fund(11061021)Supported by the Scientific Research Projection of Higher Schools of Inner Mongolia(NJZZ12011, NJ10006)+1 种基金Supported by the Program of Higher-level talents of Inner Mongolia University(125119)Supported by the Scientific Research Projection of Inner Mongolia University of Finance and Economics(KY1101)
文摘An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variables: the scalar unknown, its gradient and its flux(coefficient times the gradient), simultaneously. We also prove the existence and uniqueness of semi-discrete solution. Finally, we obtain some numerical results to illustrate the efficiency of the method.
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.
基金Supported by the Natural Science Foundation of China(Grant No.11371175)Innovation Team Project in Colleges and Universities of Guangdong Province(Grant No.2020WCXTD008)+1 种基金Science Foundation of Huashang College Guangdong University of Finance&Economics(Grant No.2020HSDS01)Science Research Team Project in Guangzhou Huashang College(Grant No.2021HSKT01).
文摘In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-up result for the semi-linear wave equation when the exponent of p is under certain conditions.Meanwhile,we derive an upper bound of the lifespan of solutions to the Cauchy problem for the semi-linear wave equation.
基金Supported by the National Natural Science Foundation of China (10471039) the Natural Science Foundation of Zhejiang Province (102009)the Natural Science Foundation of Huzhou City (2004SZX0707).
文摘A class of semi-linear Robin problem is considered. Under appropriate assumptions, the existence and asymptotic behavior of its solution are studied more carefully. Using stretched variables, the formal asymptotic expansion of solution for the problem is constructed and the uniform validity of the solution is obtained by using the method of upper and lower solution.
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In this paper we shall generalize the definition given in [1] for Lipschitz condition and contractions for functions on a non-metrizable space, besides we shall give more properties of semi-linear uniform spaces.
基金Supported by the NNSF of China(11571249)NSF of JiangSu Province(BK20171275)Supported by the grant of Innovative Training Program of College Students in Jiangsu province(201410324001Z)
文摘In this paper, by using the topological degree method and some limiting arguments, the existence of admissible periodic bouncing solutions for a class of non-conservative semi-linear impact equations is proved.
文摘The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achieving internal and boundary regional controllability. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) and Schauder’s fixed point theorem. We give a numerical example developed in internal and boundary sub region. These numerical illustrations show the efficiency of the approach and lead to conjectures.
文摘In this paper, we investigate the effect of weight function in the nonlinear part on global solvability of the Cauchy problem for a class of semi-linear hyperbolic equations with damping.
基金The Projects supported by the National Natural Science Foundation of China
文摘In this paper, by using the techniques of differential inequalities, we prove the existence of the solutions of a singularly perturbed boundary value problem for the third order semilinear differential equation with a turning point.
文摘This paper addresses the problem of identifying the unknown diffusion coefficient in a semilinear parabolic equation with incomplete initial condition.We propose an optimal control approach using the no-regret control method and the adapted low-regret control.Our approach provides a fullcharacterisation of the unknown diffusion coefficient independent of the missing initial condition.We also present an optimality system that describes the adapted low-regret control and use it to find a fuil description of the no-regret control by taking the limit of the sequence of adapted low-regret controls.
基金Project supported by the National Natural Science Foundation of China (No.10325101, No.101310310)the Science Foundation of the Ministry of Education of China (No. 20030246004).
文摘This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.
基金The work was supported by Jiangsu Province's Natural Science Foundation (BK97004)National Natural Science Foundation (19801007) of CHINA.
文摘In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation as t tends to infinity. The long asymptotic behavior of its discrete solution is obtained which is analogous to that of its continuous solution. At last, a few results are also presented for Crank-Nicolson scheme.
基金Project supported by a grant of DFG (Deutsche Forschungsgemeinschaft) for the research project "Influence of time-dependent coefficients on semi-linear wave models" (No. RE 961/17-1)
文摘The authors study the Cauchy problem for the semi-linear damped wave equation utt-△u＋b（t）ut=f（u）,u（0,χ）=u0（χ）,ut（0,χ）=u1（χ） in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b（t）〉 0 is effective, and in particular tb（t） →∞ as t →∞. The global existence of small energy data solutions for｜f（u）｜≈｜u｜^p in the supercritical case of p 〉 1＋ 2/n and p ≤n/n-2 for n ≥ 3 is proved.
基金supported by the National Natural Science Foundation of China Nos.11701283,12171482the Fundamental Research Funds for the Central Universities No.KJQN201839Science Foundation of China University of Petroleum(Beijing)No.2462020XKJS02.
文摘In this paper,we propose an immersed finite volume element method for solving the semi-linear elliptic interface problems with non-homogeneous jump conditions.Furthermore,two-grid techniques are used to improve the computational efficiency.In this way,we only need to solve a non-linear system on the coarse grid,and a linear system on the fine grid.Numerical results illustrate that the proposed method can solve the semi-linear elliptic interface problems efficiently.Approximate secondorder accuracy for the solution in the L¥norm can be obtained for the considered examples.
文摘In this paper, in the sense of the definition of almost periodicity given by H.Bohr using fixed-point principle, we investigate the existence and uniqueness of quadratic mean almost periodic solutions to semi-linear stochastic integro-differential evolution equations associated with abstract Volterra equations. Some examples are also given to illustrate our theory.
