The existence and uniqueness theorem of the screw tensor for the finite displacement of a rigidbody is proposed and then proved using the screw calculus. As a conseguence, formulae are obtained for determining the scr...The existence and uniqueness theorem of the screw tensor for the finite displacement of a rigidbody is proposed and then proved using the screw calculus. As a conseguence, formulae are obtained for determining the screw tensor in terms of the finite displacement data of the rigidbody.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 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
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
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
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 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
文摘The existence and uniqueness theorem of the screw tensor for the finite displacement of a rigidbody is proposed and then proved using the screw calculus. As a conseguence, formulae are obtained for determining the screw tensor in terms of the finite displacement data of the rigidbody.
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 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 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.
