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
Some basic problems on the level set methods were discussed, such as the method used to preserve the distance junction , the existence and uniqueness of solution for the level set equations. The main contribution is t...Some basic problems on the level set methods were discussed, such as the method used to preserve the distance junction , the existence and uniqueness of solution for the level set equations. The main contribution is to prove that in a neighborhood of the initial zero level set, the level set equations with the restriction of the distance function have a unique solution, which must be the signed distance function with respect to the evolving surface. Some skillful approaches were used: Noticing that any solution for the original equation was a distance function, the original level set equations were transformed into a simpler alternative form. Moreover, since the new system was not a classical one, the system was transformed into an ordinary one, for which the implicit function method was adopted.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>.
基金the National Natural Science Foundation of China (6001161942, 60203003)
文摘Some basic problems on the level set methods were discussed, such as the method used to preserve the distance junction , the existence and uniqueness of solution for the level set equations. The main contribution is to prove that in a neighborhood of the initial zero level set, the level set equations with the restriction of the distance function have a unique solution, which must be the signed distance function with respect to the evolving surface. Some skillful approaches were used: Noticing that any solution for the original equation was a distance function, the original level set equations were transformed into a simpler alternative form. Moreover, since the new system was not a classical one, the system was transformed into an ordinary one, for which the implicit function method was adopted.
