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 study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making s...In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making some assumptions about the nonlinear function term . The existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation are proved.展开更多
In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, ...In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space <em>E</em><sub>0</sub> and <em>E<sub>k</sub></em>, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set <em>B</em><sub><em>0k</em> </sub>on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup <em>S</em>(<em>t</em>) of the equation has the family of the global attractor <em>A<sub>k</sub></em><sub> </sub>in space <em>E<sub>k</sub></em>. Finally, we prove that the solution semigroup <em>S</em>(<em>t</em>) is Frechet differentiable on <em>E<sub>k</sub></em> via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor <em>A<sub>k</sub></em>.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
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
In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in E<...In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in E<sub>k</sub> space is proved by prior estimation and Galerkin method;Then, 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, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors A<sub>k</sub>.展开更多
On a complete noncompact Kähler manifold M^(n)(complex dimension)with non-negative Ricci curvature and Euclidean volume growth,we prove that polynomial growth holomorphic functions of degree d has an dimension up...On a complete noncompact Kähler manifold M^(n)(complex dimension)with non-negative Ricci curvature and Euclidean volume growth,we prove that polynomial growth holomorphic functions of degree d has an dimension upper bound cdn,where c depends only on n and the asymptotic volume ratio.Note that the power is sharp.展开更多
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 study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making some assumptions about the nonlinear function term . The existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation are proved.
文摘In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space <em>E</em><sub>0</sub> and <em>E<sub>k</sub></em>, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set <em>B</em><sub><em>0k</em> </sub>on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup <em>S</em>(<em>t</em>) of the equation has the family of the global attractor <em>A<sub>k</sub></em><sub> </sub>in space <em>E<sub>k</sub></em>. Finally, we prove that the solution semigroup <em>S</em>(<em>t</em>) is Frechet differentiable on <em>E<sub>k</sub></em> via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor <em>A<sub>k</sub></em>.
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.
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In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in E<sub>k</sub> space is proved by prior estimation and Galerkin method;Then, 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, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors A<sub>k</sub>.
基金Finally,he thanks the anonymous referees for improving the readability of the paper.The author was partially supported by NSFC No.12071140Program of Shanghai Academic/Technology Research Leader No.20XD1401500and the Science and Technology Commission of Shanghai Municipality No.18dz2271000,as well as the Xplore Prize by Tencent.
文摘On a complete noncompact Kähler manifold M^(n)(complex dimension)with non-negative Ricci curvature and Euclidean volume growth,we prove that polynomial growth holomorphic functions of degree d has an dimension upper bound cdn,where c depends only on n and the asymptotic volume ratio.Note that the power is sharp.