The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in whic...The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.展开更多
In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one wea...In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.展开更多
In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.
We study the well-posedness of the second order degenerate integro-differential equations （P2）： （Mu）t＇（t） ＋ a（Mu）＇（t） = Au（t） ＋ ft_c~ a（t - s）Au（s）ds ＋ f（t）, 0 ≤ t ≤ 27r, with periodic bounda...We study the well-posedness of the second order degenerate integro-differential equations （P2）： （Mu）t＇（t） ＋ a（Mu）＇（t） = Au（t） ＋ ft_c~ a（t - s）Au（s）ds ＋ f（t）, 0 ≤ t ≤ 27r, with periodic boundary conditions Mu（O） = Mu（27r）, （Mu）＇（O） = （Mu）＇（2π）, in periodic Lebesgue-Bochner spaces LP（T,X）, periodic Besov spaces BBp,q（T, X） and periodic Triebel-Lizorkin spaces F~,q（＇F, X）, where A and M are closed linear operators on a Banach space X satisfying D（A） C D（M）, a C LI（R＋） and a is a scalar number. Using known operator- valued Fourier multiplier theorems, we completely characterize the well-posedness of （P2） in the above three function spaces.展开更多
This paper is devoted to a study of the null controllability problems for one-dimensional linear degenerate wave equations through a boundary controller. First, the well-posedness of linear degenerate wave equations i...This paper is devoted to a study of the null controllability problems for one-dimensional linear degenerate wave equations through a boundary controller. First, the well-posedness of linear degenerate wave equations is discussed. Then the null controllability of some degenerate wave equations is established, when a control acts on the non-degenerate boundary. Different from the known controllability results in the case that a control acts on the degenerate boundary, any initial value in state space is controllable in this case. Also, an explicit expression for the controllability time is given. Furthermore, a counterexample on the controllability is given for some other degenerate wave equations.展开更多
In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regul...In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.展开更多
Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinan...Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent "phase shifts" of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.展开更多
In this paper,the authors mainly consider the exact controllability for degenerate wave equation,which degenerates at the interior point,and boundary controls acting at only one of the boundary points.The main results...In this paper,the authors mainly consider the exact controllability for degenerate wave equation,which degenerates at the interior point,and boundary controls acting at only one of the boundary points.The main results are that,it is possible to control both the position and the velocity at every point of the body and at a certain time T for the wave equation with interior weakly degeneracy.Moreover,it is shown that the exact controllability fails for the wave equation with interior strongly degeneracy.In order to steer the system to a certain state,one needs controls to act on both boundary points for the wave equation with interior strongly degeneracy.The difficulties are addressed by means of spectral analysis.展开更多
The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Cle...The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ＇（.） is permitted to have zero measure. This is an answer to an open problem in [13, p. 288].展开更多
This paper deals with the properties of the solution to a class of nonlocal degenerate reaction-diffusion equation with nonlocal source,subject to the null Dirichlet boundary condition.We first give sufficient conditi...This paper deals with the properties of the solution to a class of nonlocal degenerate reaction-diffusion equation with nonlocal source,subject to the null Dirichlet boundary condition.We first give sufficient conditions for that the solution exists globally or blows up in the finite time.Then the blow-up time is also given.At last,we obtain a property differing from the local source which the blow-up set is the entire interval.展开更多
In this paper, we prove the existence and nonuniqueness of the weak solutions of the initial and boundary value problem for a nonlinear degenerate parabolic equation not in divergence form. Localization property of we...In this paper, we prove the existence and nonuniqueness of the weak solutions of the initial and boundary value problem for a nonlinear degenerate parabolic equation not in divergence form. Localization property of weak solutions will be also discussed.展开更多
We give a sufficient condition for uniqueness for the pressure/saturation system. We establish this condition through analytic arguments, and then construct 'mobilities' (or mobility-like functions) that satis...We give a sufficient condition for uniqueness for the pressure/saturation system. We establish this condition through analytic arguments, and then construct 'mobilities' (or mobility-like functions) that satisfy the new condition (when the parameter is 2). For the constructed 'mobilities', we do graphical experiments that show, empirically, that this condition could be satisfied for other values of . These empirical experiments indicate that the usual smoothness condition on the fractional flow function (and on the total mobility), for uniqueness and convergence, might not be necessary. This condition is also sufficient for the convergence of a family of perturbed problems to the original pressure/saturation problem.展开更多
In this papaer, existence results for degenerate parabolic boundary valueproblems of second order are proved. The weak solution is sought in a suitable weightedSobolev space by using the generalized degree theory.
In this paper we consider the Dirichlet problems of a non-uniformly digenerate elliptic equations of the formwhose prototype iswhere n C 1 N is a bounded domain,0<b(x)<1,0<o<P.We establish that if Aand B a...In this paper we consider the Dirichlet problems of a non-uniformly digenerate elliptic equations of the formwhose prototype iswhere n C 1 N is a bounded domain,0<b(x)<1,0<o<P.We establish that if Aand B are under some structure conditions and 0<or S P<max{aam +k,o+1}tthen there exists a CI+o-solution of(0.1)associated with the Dirichlet boundary dsta.展开更多
A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole...A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole region. The integral method is used to investigate the blowup properties of the solution.展开更多
In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.
基金supported by the National Natural Science Foundation of China(Nos.11061018,11261029)the Youth Foundation of Lanzhou Jiaotong University(No.2011028)+1 种基金the Long Yuan Young Creative Talents Support Program(No.252003)the Joint Funds of the Gansu Provincial Natural Science Foundation of China(No.1212RJZA043)
文摘The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.
基金This research is supported by the Natural science Foundation of Hunan province
文摘In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.
基金Supported by the funds of the State Educational Commission of China for returned scholars from abroad
文摘In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.
基金supported by National Natural Science Foundation of China(Grant No.11171172)
文摘We study the well-posedness of the second order degenerate integro-differential equations （P2）： （Mu）t＇（t） ＋ a（Mu）＇（t） = Au（t） ＋ ft_c~ a（t - s）Au（s）ds ＋ f（t）, 0 ≤ t ≤ 27r, with periodic boundary conditions Mu（O） = Mu（27r）, （Mu）＇（O） = （Mu）＇（2π）, in periodic Lebesgue-Bochner spaces LP（T,X）, periodic Besov spaces BBp,q（T, X） and periodic Triebel-Lizorkin spaces F~,q（＇F, X）, where A and M are closed linear operators on a Banach space X satisfying D（A） C D（M）, a C LI（R＋） and a is a scalar number. Using known operator- valued Fourier multiplier theorems, we completely characterize the well-posedness of （P2） in the above three function spaces.
基金supported by the National Natural Science Foundation of China under Grant Nos.11371084,11471070 and 11171060the Fundamental Research Funds for the Central Universities under Grant Nos.14ZZ2222 and 2412015BJ011+1 种基金the National Basic Research Program of China(973 Program)under Grant No.2011CB808002the Fok Ying Tong Education Foundation under Grant No.141001
文摘This paper is devoted to a study of the null controllability problems for one-dimensional linear degenerate wave equations through a boundary controller. First, the well-posedness of linear degenerate wave equations is discussed. Then the null controllability of some degenerate wave equations is established, when a control acts on the non-degenerate boundary. Different from the known controllability results in the case that a control acts on the degenerate boundary, any initial value in state space is controllable in this case. Also, an explicit expression for the controllability time is given. Furthermore, a counterexample on the controllability is given for some other degenerate wave equations.
文摘In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11775121 and 11435005)the K. C. Wong Magna Fund in Ningbo University.
文摘Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent "phase shifts" of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.
基金supported by the National Natural Science Foundation of China under Grant No.12271316the National Natural Science Foundation of China for the Youth under Grant No.11801339+1 种基金Shanxi Sciences Project for Selected Overseas Scholars under Grant No.2018–172the Technical Innovation Team of Jinzhong University under Grant No.202111。
文摘In this paper,the authors mainly consider the exact controllability for degenerate wave equation,which degenerates at the interior point,and boundary controls acting at only one of the boundary points.The main results are that,it is possible to control both the position and the velocity at every point of the body and at a certain time T for the wave equation with interior weakly degeneracy.Moreover,it is shown that the exact controllability fails for the wave equation with interior strongly degeneracy.In order to steer the system to a certain state,one needs controls to act on both boundary points for the wave equation with interior strongly degeneracy.The difficulties are addressed by means of spectral analysis.
基金Project supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE(No.[2000]26)the 973 Project of the Ministry of Science and Technology of China(No.2006CB805902)+1 种基金the National Natural Science Foundation of China(No.10571072)the Key Laboratory of Symbolic Computation and Knowledge Engineering of the Ministry of Education of China and the 985 Project of Jilin University.
文摘The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ＇（.） is permitted to have zero measure. This is an answer to an open problem in [13, p. 288].
基金Supported by the National Natural Science Foundation of China（10571024）
文摘This paper deals with the properties of the solution to a class of nonlocal degenerate reaction-diffusion equation with nonlocal source,subject to the null Dirichlet boundary condition.We first give sufficient conditions for that the solution exists globally or blows up in the finite time.Then the blow-up time is also given.At last,we obtain a property differing from the local source which the blow-up set is the entire interval.
文摘In this paper, we prove the existence and nonuniqueness of the weak solutions of the initial and boundary value problem for a nonlinear degenerate parabolic equation not in divergence form. Localization property of weak solutions will be also discussed.
文摘We give a sufficient condition for uniqueness for the pressure/saturation system. We establish this condition through analytic arguments, and then construct 'mobilities' (or mobility-like functions) that satisfy the new condition (when the parameter is 2). For the constructed 'mobilities', we do graphical experiments that show, empirically, that this condition could be satisfied for other values of . These empirical experiments indicate that the usual smoothness condition on the fractional flow function (and on the total mobility), for uniqueness and convergence, might not be necessary. This condition is also sufficient for the convergence of a family of perturbed problems to the original pressure/saturation problem.
文摘In this papaer, existence results for degenerate parabolic boundary valueproblems of second order are proved. The weak solution is sought in a suitable weightedSobolev space by using the generalized degree theory.
文摘In this paper we consider the Dirichlet problems of a non-uniformly digenerate elliptic equations of the formwhose prototype iswhere n C 1 N is a bounded domain,0<b(x)<1,0<o<P.We establish that if Aand B are under some structure conditions and 0<or S P<max{aam +k,o+1}tthen there exists a CI+o-solution of(0.1)associated with the Dirichlet boundary dsta.
文摘A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole region. The integral method is used to investigate the blowup properties of the solution.
基金The NSFC (10626024) of ChinaChina Postdoctoral Science Foundation and Graduate Innovation Lab of Jilin University
文摘In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.
