In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(...In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(Δt 4+Δx 4) It can be easily solved by double sweeping method.展开更多
This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian,which could be proposed as a model for the epitaxial growth of thin films.By using th...This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian,which could be proposed as a model for the epitaxial growth of thin films.By using the variational method and the logarithmic type Sobolev inequality,we give some threshold results for blow-up solutions and global solutions,which could be classified by the initial energy.The asymptotic estimates about blow-up time and decay estimate of weak solutions are obtained.展开更多
In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite ...In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite time under some assumptions on the density functions.展开更多
The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p...The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.展开更多
We study the existence of solutions to the following parabolic equation{ut-△pu=λ/｜x｜s｜u｜q-2u,（x,t）∈Ω×（0,∞）,u（x,0）=f（x）,x∈Ω,u（x,t）=0,（x,t）∈Ω×（0,∞）,（P）}where-△pu ≡-div（｜▽u｜...We study the existence of solutions to the following parabolic equation{ut-△pu=λ/｜x｜s｜u｜q-2u,（x,t）∈Ω×（0,∞）,u（x,0）=f（x）,x∈Ω,u（x,t）=0,（x,t）∈Ω×（0,∞）,（P）}where-△pu ≡-div（｜▽u｜p-2▽u）,1〈p〈N,0〈s≤p,p≤q≤p＊（s） = N-s/N-pp,Ω is a bounded domain in RN such that 0∈Ω with a C1 boundaryΩ,f≥0 satisfying some convenient regularity assumptions.The analysis reveals that the existence of solutions for（P） depends on p,q,s in general,and on the relation between λ and the best constant in the Sobolev-Hardy inequality.展开更多
This paper is concerned with periodic optimal control problems governed by semi- linear parabolic differential equations with impulse control. Pontryagin＇s maximum principle is derived. The proofs rely on a unique co...This paper is concerned with periodic optimal control problems governed by semi- linear parabolic differential equations with impulse control. Pontryagin＇s maximum principle is derived. The proofs rely on a unique continuation estimate at one time for a linear parabolic equation.展开更多
We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, ...We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.展开更多
For a strongly degenerate parabolic equation, we first use the results of Ph. Blanc[1] to get that the Neumann problem may not have a global classical solution. The reason is that the gradients of some local classical...For a strongly degenerate parabolic equation, we first use the results of Ph. Blanc[1] to get that the Neumann problem may not have a global classical solution. The reason is that the gradients of some local classical solutions blow up in finite time. And under some conditions, we get the global integral solution by the use of semigroup. Then, we give a sufficient condition under which the Neumann problem has a global classical solution. Finally. we study the asymptotic behaviour of the classical solution and prove that the solution converges to the mean of the initial value.展开更多
In this paper,we consider a semilinear parabolic equation with a general nonlinearity.We establish a new finite time blow-up criterion and also derive the upper bound for the blow-up time.The results partially general...In this paper,we consider a semilinear parabolic equation with a general nonlinearity.We establish a new finite time blow-up criterion and also derive the upper bound for the blow-up time.The results partially generalize some recent ones obtained by He Ma et al.展开更多
This paper presents an explicit difference scheme with accuracy and branching stability for solving onedimensional parabolic type equation by the method of undetermined parameters and its truncation error is O(△t4+△...This paper presents an explicit difference scheme with accuracy and branching stability for solving onedimensional parabolic type equation by the method of undetermined parameters and its truncation error is O(△t4+△x4). The stability condition is r=a△t/△x2<1/2.展开更多
Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem...To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Two oscillatory criteria of solutions for systems of parabolic differential equations with deviating arguments are obtained.展开更多
Parabolic equation （PE） method is an efficient tool for modelling underwater sound propagation, particularly for problems involving range dependence. Since the PE method was first introduced into the field of underw...Parabolic equation （PE） method is an efficient tool for modelling underwater sound propagation, particularly for problems involving range dependence. Since the PE method was first introduced into the field of underwater acoustics, it has been about 40 years, during which contributions to extending its capability has been continuously made. The most recent review paper surveyed the contributions made before 1999. In the period of 2000-2016, the development of PE method basically focuses on seismo-acoustic problems, three-dimensional problems, and realistic applications. In this paper, a review covering the contribution from 2000 to 2016 is given, and what should be done in future work is also discussed.展开更多
Two dimensional parabolic stability equations (PSE) are numerically solved using expansions in orthogonal functions in the normal direction.The Chebyshev polynomials approximation,which is a very useful form of ortho...Two dimensional parabolic stability equations (PSE) are numerically solved using expansions in orthogonal functions in the normal direction.The Chebyshev polynomials approximation,which is a very useful form of orthogonal expansions, is applied to solving parabolic stability equations. It is shown that results of great accuracy are effectively obtained.The availability of using Chebyshev approximations in parabolic stability equations is confirmed.展开更多
An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary conditio...An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary condition is given for the solutions of the parameter estimatioll problem.展开更多
Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up p...Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up phenomenons for ut = ？u ＋ f（u） are discussed in terms of two cases：（i） an initial condition is given;（ii） a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.展开更多
In this paper, the Cauchy problem of the degenerate parabolic equationsis studied for some cases, and the explicit Holder estimates of the solution u with respectto x is given.
The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div（（｜u｜^δ ＋ d0）｜↓△｜^p（x,t）-2↓△u） ＋ f（x, t） （0 〈 δ 〈 2）. They a...The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div（（｜u｜^δ ＋ d0）｜↓△｜^p（x,t）-2↓△u） ＋ f（x, t） （0 〈 δ 〈 2）. They apply the method of parabolic regularization and Galerkin＇s method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L^2 （Ω） norm as t →∞.展开更多
In this paper, we extend the applications of proper orthogonal decomposition （POD） method, i.e., apply POD method to a mixed finite element （MFE） formulation naturally satisfied Brezz-Babu^ka for parabolic equatio...In this paper, we extend the applications of proper orthogonal decomposition （POD） method, i.e., apply POD method to a mixed finite element （MFE） formulation naturally satisfied Brezz-Babu^ka for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis- tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.展开更多
In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,...In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.展开更多
文摘In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(Δt 4+Δx 4) It can be easily solved by double sweeping method.
基金Supported by Shandong Provincial Natural Science Foundation of China(Grant No.ZR2021MA003).
文摘This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian,which could be proposed as a model for the epitaxial growth of thin films.By using the variational method and the logarithmic type Sobolev inequality,we give some threshold results for blow-up solutions and global solutions,which could be classified by the initial energy.The asymptotic estimates about blow-up time and decay estimate of weak solutions are obtained.
基金This work is supported in part by NNSF of China(10571126)in part by Program for New Century Excellent Talents in University
文摘In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite time under some assumptions on the density functions.
基金Yachun Li’s research was supported partly by National Natural Science Foundation of China (10571120,10971135)the Program for New Century Excellent Talents of Chinese Ministry of Education (NCET-07-0546)+3 种基金Shanghai Shuguang Project 06SG11Zhigang Wang’s research was supported partly by Shanghai Jiao Tong University Innovation Fund For Postgraduates (AE071202)the University Young Teacher Sciences Foundation of Anhui Province (2010SQRL145)the Quality Project Found of Fuyang Normal College (2010JPKC07)
文摘The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.
基金supported by NPU Foundation for Fundamental Research (NPU-FFR-JC201124)NSF of China (10871157,11001221,11002110)Specialized Research Fund for the Doctoral Program in Higher Education (200806990032)
文摘We study the existence of solutions to the following parabolic equation{ut-△pu=λ/｜x｜s｜u｜q-2u,（x,t）∈Ω×（0,∞）,u（x,0）=f（x）,x∈Ω,u（x,t）=0,（x,t）∈Ω×（0,∞）,（P）}where-△pu ≡-div（｜▽u｜p-2▽u）,1〈p〈N,0〈s≤p,p≤q≤p＊（s） = N-s/N-pp,Ω is a bounded domain in RN such that 0∈Ω with a C1 boundaryΩ,f≥0 satisfying some convenient regularity assumptions.The analysis reveals that the existence of solutions for（P） depends on p,q,s in general,and on the relation between λ and the best constant in the Sobolev-Hardy inequality.
基金partially supported by the National Science Foundation of China(11371285)
文摘This paper is concerned with periodic optimal control problems governed by semi- linear parabolic differential equations with impulse control. Pontryagin＇s maximum principle is derived. The proofs rely on a unique continuation estimate at one time for a linear parabolic equation.
基金Project supported by the National Natural Science Foundation of China(Grant No.11471262)the National Basic Research Program of China(Grant No.2012CB025904)the State Key Laboratory of Science and Engineering Computing and the Center for High Performance Computing of Northwestern Polytechnical University,China
文摘We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.
文摘For a strongly degenerate parabolic equation, we first use the results of Ph. Blanc[1] to get that the Neumann problem may not have a global classical solution. The reason is that the gradients of some local classical solutions blow up in finite time. And under some conditions, we get the global integral solution by the use of semigroup. Then, we give a sufficient condition under which the Neumann problem has a global classical solution. Finally. we study the asymptotic behaviour of the classical solution and prove that the solution converges to the mean of the initial value.
基金Supported by the Nation Natural Science Foundation of China(Grant No.11271141)Chongqing Science and Technology Commission(Grant No.cstc2018jcyjAX0787).
文摘In this paper,we consider a semilinear parabolic equation with a general nonlinearity.We establish a new finite time blow-up criterion and also derive the upper bound for the blow-up time.The results partially generalize some recent ones obtained by He Ma et al.
文摘This paper presents an explicit difference scheme with accuracy and branching stability for solving onedimensional parabolic type equation by the method of undetermined parameters and its truncation error is O(△t4+△x4). The stability condition is r=a△t/△x2<1/2.
文摘Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
文摘To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Two oscillatory criteria of solutions for systems of parabolic differential equations with deviating arguments are obtained.
基金Project supported by the Foundation of State Key Laboratory of Acoustics,Institute of Acoustics,Chinese Academy of Sciences(Grant No.SKLA201303)the National Natural Science Foundation of China(Grant Nos.11104044,11234002,and 11474073)
文摘Parabolic equation （PE） method is an efficient tool for modelling underwater sound propagation, particularly for problems involving range dependence. Since the PE method was first introduced into the field of underwater acoustics, it has been about 40 years, during which contributions to extending its capability has been continuously made. The most recent review paper surveyed the contributions made before 1999. In the period of 2000-2016, the development of PE method basically focuses on seismo-acoustic problems, three-dimensional problems, and realistic applications. In this paper, a review covering the contribution from 2000 to 2016 is given, and what should be done in future work is also discussed.
文摘Two dimensional parabolic stability equations (PSE) are numerically solved using expansions in orthogonal functions in the normal direction.The Chebyshev polynomials approximation,which is a very useful form of orthogonal expansions, is applied to solving parabolic stability equations. It is shown that results of great accuracy are effectively obtained.The availability of using Chebyshev approximations in parabolic stability equations is confirmed.
基金the post-doctoral funds of China and funds of State Educational Commission of China for returned scholars from abroad
文摘An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary condition is given for the solutions of the parameter estimatioll problem.
基金supported by the National Science Foundation of China(11671401)supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(17XNH106)
文摘Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up phenomenons for ut = ？u ＋ f（u） are discussed in terms of two cases：（i） an initial condition is given;（ii） a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.
文摘In this paper, the Cauchy problem of the degenerate parabolic equationsis studied for some cases, and the explicit Holder estimates of the solution u with respectto x is given.
基金Supported by NSFC (10771085)Graduate Innovation Fund of Jilin University(20111034)the 985 program of Jilin University
文摘The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div（（｜u｜^δ ＋ d0）｜↓△｜^p（x,t）-2↓△u） ＋ f（x, t） （0 〈 δ 〈 2）. They apply the method of parabolic regularization and Galerkin＇s method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L^2 （Ω） norm as t →∞.
基金supported by the National Science Foundation of China(11271127 and 11061009)Science Research Program of Guizhou(GJ[2011]2367)the Co-Construction Project of Beijing Municipal Commission of Education
文摘In this paper, we extend the applications of proper orthogonal decomposition （POD） method, i.e., apply POD method to a mixed finite element （MFE） formulation naturally satisfied Brezz-Babu^ka for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis- tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.
基金supported by the National Science Foundation of China(41275063 and 11401575)
文摘In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.
