This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utiliz...This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.展开更多
In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations(?-?/(?t))u(x, t) + h(x,t)u(x,t) = 0 and nonlinear parabolic equations(?-?-/(?t))u(x,t) + h(x,...In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations(?-?/(?t))u(x, t) + h(x,t)u(x,t) = 0 and nonlinear parabolic equations(?-?-/(?t))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
We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div(σ(x)▽u) and multiplicative noises. Under some mild conditions on the diffusion variable σ(x) and with...We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div(σ(x)▽u) and multiplicative noises. Under some mild conditions on the diffusion variable σ(x) and without any restriction on the upper growth p of nonlinearity, except that p > 2, we show the existences of random attractor in D_0^(1,2)(D_N, σ) (■∈ [2, 2p-2]) space, where D_N is an arbitrary(bounded or unbounded) domain in R^N, N ≥ 2. For this purpose, some abstract results based on the omega-limit compactness are established.展开更多
This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite differenc...This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.展开更多
Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are se...Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are second order correct in time.展开更多
In this paper,we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations ut u=g(u)+λf(x),(x,t)×(0,T),u=0,(x,t)∈×[0,T),u(x,0)=u0(x)≥0,x ∈.(P) By...In this paper,we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations ut u=g(u)+λf(x),(x,t)×(0,T),u=0,(x,t)∈×[0,T),u(x,0)=u0(x)≥0,x ∈.(P) By combining a priori estimate of global solution with property of stationary solution set of problem(P),we prove that the minimal stationary solution Uλ(x)of problem(P)is stable,whereas,any other stationary solution is an initial datum threshold for the existence and nonexistence of global solution to problem(P).展开更多
In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner t...In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family.展开更多
In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blo...In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.展开更多
Models based on a parabolic equation(PE)can accurately predict sound propagation problems in range-dependent ocean waveguides.Consequently,this method has developed rapidly in recent years.Compared with normal mode th...Models based on a parabolic equation(PE)can accurately predict sound propagation problems in range-dependent ocean waveguides.Consequently,this method has developed rapidly in recent years.Compared with normal mode theory,PE focuses on numerical calculation,which is difficult to use in the mode domain analysis of sound propagation,such as the calculation of mode phase velocity and group velocity.To broaden the capability of PE models in analyzing the underwater sound field,a wave mode calculation method based on PE is proposed in this study.Step-split Pade PE recursive matrix equations are combined to obtain a propagation matrix.Then,the eigenvalue decomposition technique is applied to the matrix to extract sound mode eigenvalues and eigenfunctions.Numerical experiments on some typical waveguides are performed to test the accuracy and flexibility of the new method.Discussions on different orders of Padéapproximant demonstrate angle limitations in PE and the missing root problem is also discussed to prove the advantage of the new method.The PE mode method can be expanded in the future to solve smooth wave modes in ocean waveguides,including fluctuating boundaries and sound speed profiles.展开更多
The existences of the global attractor Ae for a degenerate parabolic equation and of the homogenized attractor A0 for the corresponding homogenized equation are studied, and explicit estimates for the distance between...The existences of the global attractor Ae for a degenerate parabolic equation and of the homogenized attractor A0 for the corresponding homogenized equation are studied, and explicit estimates for the distance between Ae and A0 are given.展开更多
In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the d...In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the delay term f and the non-autonomous forcing term g,we prove the existence of uniform attractors in Banach space CH1(RN)for the multivalued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.展开更多
In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are ...In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.展开更多
We analyze the discretization of a Neumann boundary control problem with a stochastic parabolic equation, where additive noise occurs in the Neumann boundary condition. The convergence is established for general filtr...We analyze the discretization of a Neumann boundary control problem with a stochastic parabolic equation, where additive noise occurs in the Neumann boundary condition. The convergence is established for general filtration, and the convergence rate O(τ1/4-?+ h1/2-?) is derived for the natural filtration of the Q-Wiener process.展开更多
This paper is mainly focused on the global existence and extinction behaviour of the solutions to a doubly nonlinear parabolic equation with logarithmic nonlinearity. By making use of energy estimates method and a ser...This paper is mainly focused on the global existence and extinction behaviour of the solutions to a doubly nonlinear parabolic equation with logarithmic nonlinearity. By making use of energy estimates method and a series of ordinary differential inequalities, the global existence of the solution is obtained. Moreover, we give the sufficient conditions on the occurrence(or absence)of the extinction behaviour.展开更多
THIS letter discusses the following initial-boundary value problem: where (?)Ω is the boundary of domain Ω, f(s): [0, b)→(0, +∞) satisfy Because this problem has strong background of physics and geometry, it has a...THIS letter discusses the following initial-boundary value problem: where (?)Ω is the boundary of domain Ω, f(s): [0, b)→(0, +∞) satisfy Because this problem has strong background of physics and geometry, it has attracted much attention in recent years. The main results can be found in ref. [1]. Here we give only a result of reference [2].展开更多
It is important for the wireless communication field to conduct research on large-scale complex electromagnetic environment(CEME)simulation.There exist many models for computing CEME simulation,including empirical mod...It is important for the wireless communication field to conduct research on large-scale complex electromagnetic environment(CEME)simulation.There exist many models for computing CEME simulation,including empirical models,half-empirical or halfdeterministic models and deterministic models.Most of these models cannot obtain satisfactory results due to the limitation of the capacity of computers.The ray tracing(RT)and parabolic equation(PE)methods are very suitable for large-scale CEME simulation.Based on the introduction of RT and PE,qualitative comparisons of the two methods are analyzed in view of algorithm theory,the category of the model,solution to the model and the application field,and then four specific indices are focused on to analyze the computational complexity,accuracy,speed and parallelism in details.The numerical experiments are presented by the three-dimensional(3D)RT method employing the software of Wireless InSite(WI)and a quasi-3DPE method using the sliced method.Although both RT and PE methods can achieve high speedup using coarse-grained parallel computing,the experimental results indicate that the PE method can obtain a higher speed than the RT method,and the two methods can acquire an approximate precision.A hybrid procedure using both RT and PE methods can obtain a better result for solving CEME problems.展开更多
In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the ...In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).展开更多
In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and co...In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial discretization.The method is based on a new cellcentered meshes,and it is locally conservative.It has smaller truncation error than the classical finite volume method on uniform meshes.We use the framework of the gradient discretization method to analyze the stability and convergence.The numerical experiments show that the new method has second-order convergence.Moreover,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.展开更多
A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation.This method is considered whe...A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation.This method is considered where the nonlinear problem is solved only on a coarse grid of size H and two linear problems based on the coarse-grid solutions and one Newton iteration is considered on a fine grid of size h.We provide the rigorous error estimate,which demonstrates that our scheme converges with order O(∆t^(2−α)+h^(2)+H^(4))on non-uniform rectangular grid.This result indicates that the improved two-grid method can obtain asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H^(2)).Finally,numerical tests confirm the theoretical results of the presented method.展开更多
文摘This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.
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(?-?/(?t))u(x, t) + h(x,t)u(x,t) = 0 and nonlinear parabolic equations(?-?-/(?t))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).
基金supported by China NSF(11271388)Scientific and Technological Research Program of Chongqing Municipal Education Commission(KJ1400430)Basis and Frontier Research Project of Chongqing(cstc2014jcyj A00035)
文摘We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div(σ(x)▽u) and multiplicative noises. Under some mild conditions on the diffusion variable σ(x) and without any restriction on the upper growth p of nonlinearity, except that p > 2, we show the existences of random attractor in D_0^(1,2)(D_N, σ) (■∈ [2, 2p-2]) space, where D_N is an arbitrary(bounded or unbounded) domain in R^N, N ≥ 2. For this purpose, some abstract results based on the omega-limit compactness are established.
文摘This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.
基金National Natural Science Foundation and Doctoral Foundation of Education Ministry of Stat
文摘Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are second order correct in time.
基金supported by Natural Science Foundation of China(10971061)Hunan Provincial Innovation Foundation For Postgraduate(CX2010B209)
文摘In this paper,we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations ut u=g(u)+λf(x),(x,t)×(0,T),u=0,(x,t)∈×[0,T),u(x,0)=u0(x)≥0,x ∈.(P) By combining a priori estimate of global solution with property of stationary solution set of problem(P),we prove that the minimal stationary solution Uλ(x)of problem(P)is stable,whereas,any other stationary solution is an initial datum threshold for the existence and nonexistence of global solution to problem(P).
文摘In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family.
基金Project supported by the Youth Foundation of the National Natural Science Foundation of China(No. 10701061)
文摘In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.
基金Project supported by Young Elite Scientist Sponsorship Program by CAST(Grant No.YESS20200330).
文摘Models based on a parabolic equation(PE)can accurately predict sound propagation problems in range-dependent ocean waveguides.Consequently,this method has developed rapidly in recent years.Compared with normal mode theory,PE focuses on numerical calculation,which is difficult to use in the mode domain analysis of sound propagation,such as the calculation of mode phase velocity and group velocity.To broaden the capability of PE models in analyzing the underwater sound field,a wave mode calculation method based on PE is proposed in this study.Step-split Pade PE recursive matrix equations are combined to obtain a propagation matrix.Then,the eigenvalue decomposition technique is applied to the matrix to extract sound mode eigenvalues and eigenfunctions.Numerical experiments on some typical waveguides are performed to test the accuracy and flexibility of the new method.Discussions on different orders of Padéapproximant demonstrate angle limitations in PE and the missing root problem is also discussed to prove the advantage of the new method.The PE mode method can be expanded in the future to solve smooth wave modes in ocean waveguides,including fluctuating boundaries and sound speed profiles.
基金the Scientific Research Foundation for Returned Overseas Chinese Scholars under the State Education Ministrythe Key Teachers’Foundation of Chongqing University
文摘The existences of the global attractor Ae for a degenerate parabolic equation and of the homogenized attractor A0 for the corresponding homogenized equation are studied, and explicit estimates for the distance between Ae and A0 are given.
基金Supported by the 2018 research funding of higher education of Gansu province project[2018B-075]
文摘In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the delay term f and the non-autonomous forcing term g,we prove the existence of uniform attractors in Banach space CH1(RN)for the multivalued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.
基金supported by the National Natural Science Foundation of China(No.12271039)the Natural Science Foundation of universities of Anhui Province of China(Grant Nos.KJ2021A0927,2023AH040161).
文摘In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.
基金supported by National Natural Science Foundation of China (Grant No.11901410)the Fundamental Research Funds for the Central Universities in China (Grant No. 2020SCU12063)。
文摘We analyze the discretization of a Neumann boundary control problem with a stochastic parabolic equation, where additive noise occurs in the Neumann boundary condition. The convergence is established for general filtration, and the convergence rate O(τ1/4-?+ h1/2-?) is derived for the natural filtration of the Q-Wiener process.
基金Supported by the Project of Education Department of Hunan Province (20A174)。
文摘This paper is mainly focused on the global existence and extinction behaviour of the solutions to a doubly nonlinear parabolic equation with logarithmic nonlinearity. By making use of energy estimates method and a series of ordinary differential inequalities, the global existence of the solution is obtained. Moreover, we give the sufficient conditions on the occurrence(or absence)of the extinction behaviour.
文摘THIS letter discusses the following initial-boundary value problem: where (?)Ω is the boundary of domain Ω, f(s): [0, b)→(0, +∞) satisfy Because this problem has strong background of physics and geometry, it has attracted much attention in recent years. The main results can be found in ref. [1]. Here we give only a result of reference [2].
文摘It is important for the wireless communication field to conduct research on large-scale complex electromagnetic environment(CEME)simulation.There exist many models for computing CEME simulation,including empirical models,half-empirical or halfdeterministic models and deterministic models.Most of these models cannot obtain satisfactory results due to the limitation of the capacity of computers.The ray tracing(RT)and parabolic equation(PE)methods are very suitable for large-scale CEME simulation.Based on the introduction of RT and PE,qualitative comparisons of the two methods are analyzed in view of algorithm theory,the category of the model,solution to the model and the application field,and then four specific indices are focused on to analyze the computational complexity,accuracy,speed and parallelism in details.The numerical experiments are presented by the three-dimensional(3D)RT method employing the software of Wireless InSite(WI)and a quasi-3DPE method using the sliced method.Although both RT and PE methods can achieve high speedup using coarse-grained parallel computing,the experimental results indicate that the PE method can obtain a higher speed than the RT method,and the two methods can acquire an approximate precision.A hybrid procedure using both RT and PE methods can obtain a better result for solving CEME problems.
基金supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China 10971074+1 种基金the National Basic Research Program under the Grant 2005CB321703Hunan Provincial Innovation Foundation For Postgraduate CX2009B119。
文摘In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).
基金supported by the National Natural Science Foundation of China(No.11971069),NSAF(No.U1630249)and Science Challenge Project(No.TZ2016002).
文摘In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial discretization.The method is based on a new cellcentered meshes,and it is locally conservative.It has smaller truncation error than the classical finite volume method on uniform meshes.We use the framework of the gradient discretization method to analyze the stability and convergence.The numerical experiments show that the new method has second-order convergence.Moreover,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.
基金supported by the National Natural Science Foundation of China(Grant 11901489)China Postdoctoral Science Foundation(Grants BX20190187,2019M650152)+3 种基金supported by the State Key Program of National Natural Science Foundation of China(Grant 11931003)National Natural Science Foundation of China(Grants 41974133,11671157)supported by the Shandong Province Natural Science Foundation(Grant ZR2018MAQ008)National Natural Science Foundation of China(Grant 11771375).
文摘A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation.This method is considered where the nonlinear problem is solved only on a coarse grid of size H and two linear problems based on the coarse-grid solutions and one Newton iteration is considered on a fine grid of size h.We provide the rigorous error estimate,which demonstrates that our scheme converges with order O(∆t^(2−α)+h^(2)+H^(4))on non-uniform rectangular grid.This result indicates that the improved two-grid method can obtain asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H^(2)).Finally,numerical tests confirm the theoretical results of the presented method.
