In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial condition...In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality.展开更多
The initial boundary value problem for a class of high-order Beam equations with quasilinear and strongly damped terms is studied. Firstly, the existence and uniqueness of the global solution of the equation are prove...The initial boundary value problem for a class of high-order Beam equations with quasilinear and strongly damped terms is studied. Firstly, the existence and uniqueness of the global solution of the equation are proved by prior estimation and Galerkin finite element method. Then the bounded absorption set is obtained by prior estimation, and the family of global attractors for the high-order Kirchhoff-Beam equation is obtained. The Frechet differentiability of the solution semigroup is proved after the linearization of the equation, and the decay of the volume element of the linearization problem is further proved. Finally, the Hausdorff dimension and Fractal dimension of the family of global attractors are proved to be finite.展开更多
In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness o...In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.展开更多
We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make a...We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H<sub>1</sub>) - (H<sub>4</sub>). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.展开更多
In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditio...In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.展开更多
In this paper we investigate the global attractors for the one-dimensional linear model of thermodiffusion with second sound. Using the method of contractive functions, we obtain the asymptotically compact of the semi...In this paper we investigate the global attractors for the one-dimensional linear model of thermodiffusion with second sound. Using the method of contractive functions, we obtain the asymptotically compact of the semigroup and the existence of the global展开更多
This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the ove...This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the overall solution of the above problem by using a priori estimates in the spaces of E<sub>0</sub> and E<sub>k</sub>, and secondly, it proves that there is a family of global attractors for the above problem, and finally estimates the Hausdorff dimension and the Fractal dimension of the family of global attractors.展开更多
In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior est...In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior estimates of the equation in E<sub>0</sub> and E<sub>k</sub> space, and then the existence and uniqueness of solution is verified by Galerkin’s method. Then, the solution semigroup S(t) is defined, and the bounded absorptive set B<sub>k</sub> is obtained on the basis of prior estimation. Through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors A<sub>k</sub> in space E<sub>k</sub>. Finally, by linearizing the equation, it is proved that the solution semigroup S(t) is Frechet differentiable on E<sub>k</sub>, and the family of global attractors A<sub>k</sub> have finite Hausdroff dimension and Fractal dimension.展开更多
The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the probl...The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.展开更多
In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded doma...In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.展开更多
In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions a...In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions are proved. At last, the existence of global attractors and its upper semicontinuous property are discussed.展开更多
In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term <span style...In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term <span style="white-space:nowrap;"><em>g</em> (<em>u</em>)</span> and Kirchhoff stress term <span style="white-space:nowrap;"><em>M</em> (<em>s</em>)</span> in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set <em>B</em><sub>0<em>k</em></sub> is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup <span style="white-space:nowrap;"><em>S</em> (<em>t</em>)</span> generated by the equation has a family of the global attractor <span style="white-space:nowrap;"><em>A</em><sub><em>k</em></sub></span> in the phase space <img src="Edit_250265b5-40f0-4b6c-b669-958eb1938010.png" width="120" height="20" alt="" />. Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on <em>E<sub>k</sub></em>. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor <em>A<sub>k</sub></em> was obtained.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 paper we establish some theorems which are concerned with the equivalent norms of Sobolev spaces on a Riemannian manifold. Using the theorems we prove the existence of global attractors for the initial value p...In this paper we establish some theorems which are concerned with the equivalent norms of Sobolev spaces on a Riemannian manifold. Using the theorems we prove the existence of global attractors for the initial value problem of Cahn-Hilliard equations. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors are also obtained.展开更多
We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of ar...We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of arbitrary order.展开更多
In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means o...In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.展开更多
In this paper we consider the Burger-Ginzburg-Landau equations, and prove the existence of the global attractor in with finite Hausdorff and fractal dimensions.
In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Mira...In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7, 8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.展开更多
In this paper, the existence of global attractors for the 2D autonomous g- Navier-Stokes equations on multi-connected bounded domains is investigated under the general assumptions of boundaries. The estimation of the ...In this paper, the existence of global attractors for the 2D autonomous g- Navier-Stokes equations on multi-connected bounded domains is investigated under the general assumptions of boundaries. The estimation of the Hausdorff dimensions for global attractors is given.展开更多
文摘In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality.
文摘The initial boundary value problem for a class of high-order Beam equations with quasilinear and strongly damped terms is studied. Firstly, the existence and uniqueness of the global solution of the equation are proved by prior estimation and Galerkin finite element method. Then the bounded absorption set is obtained by prior estimation, and the family of global attractors for the high-order Kirchhoff-Beam equation is obtained. The Frechet differentiability of the solution semigroup is proved after the linearization of the equation, and the decay of the volume element of the linearization problem is further proved. Finally, the Hausdorff dimension and Fractal dimension of the family of global attractors are proved to be finite.
文摘In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.
文摘We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H<sub>1</sub>) - (H<sub>4</sub>). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.
文摘In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.
基金Supported by the NNSF of China(11031003,11271066)Supported by the Shanghai Education Commission(13ZZ048)
文摘In this paper we investigate the global attractors for the one-dimensional linear model of thermodiffusion with second sound. Using the method of contractive functions, we obtain the asymptotically compact of the semigroup and the existence of the global
文摘This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the overall solution of the above problem by using a priori estimates in the spaces of E<sub>0</sub> and E<sub>k</sub>, and secondly, it proves that there is a family of global attractors for the above problem, and finally estimates the Hausdorff dimension and the Fractal dimension of the family of global attractors.
文摘In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior estimates of the equation in E<sub>0</sub> and E<sub>k</sub> space, and then the existence and uniqueness of solution is verified by Galerkin’s method. Then, the solution semigroup S(t) is defined, and the bounded absorptive set B<sub>k</sub> is obtained on the basis of prior estimation. Through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors A<sub>k</sub> in space E<sub>k</sub>. Finally, by linearizing the equation, it is proved that the solution semigroup S(t) is Frechet differentiable on E<sub>k</sub>, and the family of global attractors A<sub>k</sub> have finite Hausdroff dimension and Fractal dimension.
文摘The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.
文摘In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.
基金National Natural Science Foundation of China ( No. 11031003) Fund of Excellent Young Teachers in Shanghai,China( No.shgcjs008) Initial Fund of Shanghai University of Engineering Science,China( No. A-0501-11-016)
文摘In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions are proved. At last, the existence of global attractors and its upper semicontinuous property are discussed.
文摘In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term <span style="white-space:nowrap;"><em>g</em> (<em>u</em>)</span> and Kirchhoff stress term <span style="white-space:nowrap;"><em>M</em> (<em>s</em>)</span> in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set <em>B</em><sub>0<em>k</em></sub> is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup <span style="white-space:nowrap;"><em>S</em> (<em>t</em>)</span> generated by the equation has a family of the global attractor <span style="white-space:nowrap;"><em>A</em><sub><em>k</em></sub></span> in the phase space <img src="Edit_250265b5-40f0-4b6c-b669-958eb1938010.png" width="120" height="20" alt="" />. Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on <em>E<sub>k</sub></em>. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor <em>A<sub>k</sub></em> was obtained.
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 Natural Science Foundation of Henan Education Department (No.2007110004) Supported by the Natural Science Foundation of Henan University (No.06YBZR027).
文摘In this paper we establish some theorems which are concerned with the equivalent norms of Sobolev spaces on a Riemannian manifold. Using the theorems we prove the existence of global attractors for the initial value problem of Cahn-Hilliard equations. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors are also obtained.
基金Supported in part by the NSFC,Grant(10471056)Trans-Century Training Programme Foundation for the Talents of the State Education Commission
文摘We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of arbitrary order.
基金Supported by the NSF of China (Nos. 10571142, 10771167)
文摘In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.
基金the Scientific Research Foundation for Returned Overseas Chinese Scholars. State Education Commission.
文摘In this paper we consider the Burger-Ginzburg-Landau equations, and prove the existence of the global attractor in with finite Hausdorff and fractal dimensions.
基金supported by NSFC Grant (11031003)the Fundamental Research Funds for the Central Universities+1 种基金support by Fund of excellent young teachers in Shanghai (shgcjs008)Initial Fund of SUES (A-0501-11-016)
文摘In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7, 8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.
基金Project supported by the National Natural Science Fundation of China (No. 11171269)the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2012JM1012)the Scientific Research Program Funded by Shaanxi Provincial Education Department (No. 12JK0849)
文摘In this paper, the existence of global attractors for the 2D autonomous g- Navier-Stokes equations on multi-connected bounded domains is investigated under the general assumptions of boundaries. The estimation of the Hausdorff dimensions for global attractors is given.
