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
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>.
