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 note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate pa...In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.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
The perfectly matched layer(PML) is an effective technique for truncating unbounded domains with minimal spurious reflections. A fluid parabolic equation(PE) model applying PML technique was previously used to analyze...The perfectly matched layer(PML) is an effective technique for truncating unbounded domains with minimal spurious reflections. A fluid parabolic equation(PE) model applying PML technique was previously used to analyze the sound propagation problem in a range-dependent waveguide(Lu and Zhu, 2007). However, Lu and Zhu only considered a standard fluid PE to demonstrate the capability of the PML and did not take improved one-way models into consideration. They applied a [1/1] Padé approximant to the parabolic equation. The higher-order PEs are more accurate than standard ones when a very large angle propagation is considered. As for range-dependent problems, the techniques to handle the vertical interface between adjacent regions are mainly energy conserving and single-scattering. In this paper, the PML technique is generalized to the higher order elastic PE, as is to the higher order fluid PE. The correction of energy conserving is used in range-dependent waveguides. Simulation is made in both acoustic cases and seismo-acoustic cases. Range-independent and range-dependent waveguides are both adopted to test the accuracy and efficiency of this method. The numerical results illustrate that a PML is much more effective than an artificial absorbing layer(ABL) both in acoustic and seismo-acoustic sound propagation modeling.展开更多
A highly efficient H1-Galerkin mixed finite element method(MFEM) is presented with linear triangular element for the parabolic integro-differential equation.Firstly, some new results about the integral estimation and ...A highly efficient H1-Galerkin mixed finite element method(MFEM) is presented with linear triangular element for the parabolic integro-differential equation.Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O(h2) for both the original variable u in H1(π) norm and the flux p =u in H(div,π) norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O(h3) are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.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.
In this paper, the existence of solutions for discontinuous nonlinear parabolic differential IBVP is proved by using a more generalized monotone iterative method. Moreover, the convergence of this method is discussed.
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
In this paper, we study the initial-boundary value problem for the semilinear parabolic equations ut -△Xu=|u|^p-1u, where X=(X1, X2,…, Xm) is a system of real smooth vector fields which satisfy the Hormander's c...In this paper, we study the initial-boundary value problem for the semilinear parabolic equations ut -△Xu=|u|^p-1u, where X=(X1, X2,…, Xm) is a system of real smooth vector fields which satisfy the Hormander's condition, and △x=m∑j=1Xj^2is a finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.展开更多
We considerer parabolic partial differential equations under the conditions on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve...We considerer parabolic partial differential equations under the conditions on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse moments problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. Also we consider the one- dimensional one-phase inverse Stefan problem.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
In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-E′mery Ricci tensor bounded below...In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-E′mery Ricci tensor bounded below: One is u_t=Δ_fu + aulogu+bu with a, b two real constants, and another is u_t = Δ_fu+λu~α with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.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
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
A three-dimensional(3D) parabolic equation(PE) model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal d...A three-dimensional(3D) parabolic equation(PE) model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid-elastic interfaces.展开更多
The authors of this article study the existence and uniqueness of weak solutions of the initial-boundary value problem for u t = div((|u| σ + d 0)|▽u|p(x,t)-2▽u)+f(x,t)(0<σ<2).They apply the method of parabo...The authors of this article study the existence and uniqueness of weak solutions of the initial-boundary value problem for u t = div((|u| σ + d 0)|▽u|p(x,t)-2▽u)+f(x,t)(0<σ<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
A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is d...A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.展开更多
This paper puts forward a new method to solve the electromagnetic parabolic equation(EMPE)by taking the vertically-layered inhomogeneous characteristics of the atmospheric refractive index into account.First,the Fouri...This paper puts forward a new method to solve the electromagnetic parabolic equation(EMPE)by taking the vertically-layered inhomogeneous characteristics of the atmospheric refractive index into account.First,the Fourier transform and the convolution theorem are employed,and the second-order partial diferential equation,i.e.,the EMPE,in the height space is transformed into first-order constant coefcient diferential equations in the frequency space.Then,by use of the lower triangular characteristics of the coefcient matrix,the numerical solutions are designed.Through constructing analytical solutions to the EMPE,the feasibility of the new method is validated.Finally,the numerical solutions to the new method are compared with those of the commonly used split-step Fourier algorithm.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.
文摘In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.
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 Foundation of State Key Laboratory of Acoustics,Institute of Acoustics,Chinese Academy of Sciences(No.SKLA201303)the National Natural Science Foundation of China(Nos.11104044,11234002,and 11474073)
文摘The perfectly matched layer(PML) is an effective technique for truncating unbounded domains with minimal spurious reflections. A fluid parabolic equation(PE) model applying PML technique was previously used to analyze the sound propagation problem in a range-dependent waveguide(Lu and Zhu, 2007). However, Lu and Zhu only considered a standard fluid PE to demonstrate the capability of the PML and did not take improved one-way models into consideration. They applied a [1/1] Padé approximant to the parabolic equation. The higher-order PEs are more accurate than standard ones when a very large angle propagation is considered. As for range-dependent problems, the techniques to handle the vertical interface between adjacent regions are mainly energy conserving and single-scattering. In this paper, the PML technique is generalized to the higher order elastic PE, as is to the higher order fluid PE. The correction of energy conserving is used in range-dependent waveguides. Simulation is made in both acoustic cases and seismo-acoustic cases. Range-independent and range-dependent waveguides are both adopted to test the accuracy and efficiency of this method. The numerical results illustrate that a PML is much more effective than an artificial absorbing layer(ABL) both in acoustic and seismo-acoustic sound propagation modeling.
基金Project supported by the National Natural Science Foundation of China(Nos.10971203,11271340,and 11101381)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20094101110006)
文摘A highly efficient H1-Galerkin mixed finite element method(MFEM) is presented with linear triangular element for the parabolic integro-differential equation.Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O(h2) for both the original variable u in H1(π) norm and the flux p =u in H(div,π) norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O(h3) are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.
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.
文摘In this paper, the existence of solutions for discontinuous nonlinear parabolic differential IBVP is proved by using a more generalized monotone iterative method. Moreover, the convergence of this method is discussed.
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 National Natural Science Foundation of China(11631011 and 11626251)
文摘In this paper, we study the initial-boundary value problem for the semilinear parabolic equations ut -△Xu=|u|^p-1u, where X=(X1, X2,…, Xm) is a system of real smooth vector fields which satisfy the Hormander's condition, and △x=m∑j=1Xj^2is a finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.
文摘We considerer parabolic partial differential equations under the conditions on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse moments problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. Also we consider the one- dimensional one-phase inverse Stefan problem.
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In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-E′mery Ricci tensor bounded below: One is u_t=Δ_fu + aulogu+bu with a, b two real constants, and another is u_t = Δ_fu+λu~α with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.
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.
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Project supported by the National Nature Science Foundation of China(Grant Nos.11234002 and 11704337)the National Key Research Program of China(Grant No.2016YFC1400100)
文摘A three-dimensional(3D) parabolic equation(PE) model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid-elastic interfaces.
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 solutions of the initial-boundary value problem for u t = div((|u| σ + d 0)|▽u|p(x,t)-2▽u)+f(x,t)(0<σ<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 Natural Science Foundation of China (10671184)
文摘A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.
基金supported by the National Natural Science Foundation of China(Nos.41175025 and41275113)
文摘This paper puts forward a new method to solve the electromagnetic parabolic equation(EMPE)by taking the vertically-layered inhomogeneous characteristics of the atmospheric refractive index into account.First,the Fourier transform and the convolution theorem are employed,and the second-order partial diferential equation,i.e.,the EMPE,in the height space is transformed into first-order constant coefcient diferential equations in the frequency space.Then,by use of the lower triangular characteristics of the coefcient matrix,the numerical solutions are designed.Through constructing analytical solutions to the EMPE,the feasibility of the new method is validated.Finally,the numerical solutions to the new method are compared with those of the commonly used split-step Fourier algorithm.