In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations <img src="Edit_e49f9c34-0a5d-4ef2-828f-b...In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations <img src="Edit_e49f9c34-0a5d-4ef2-828f-ba3f0912bed3.png" alt="" />with strong damping terms. We will properly assume the stress term <i>M(s)</i><span style="position:relative;top:6pt;"><v:shape id="_x0000_i1026" type="#_x0000_t75" o:ole="" style="width:27pt;height:17.25pt;"><v:imagedata src="file:///C:\Users\TEST~1.SCI\AppData\Local\Temp\msohtmlclip1\01\clip_image002.wmz" o:title=""></v:imagedata></v:shape></span> and<span style="letter-spacing:-0.2pt;"> nonlinear term g(u<sub>t</sub>)<span style="position:relative;top:6pt;"><v:shape id="_x0000_i1027" type="#_x0000_t75" o:ole="" style="width:27pt;height:17.25pt;"><v:imagedata src="file:///C:\Users\TEST~1.SCI\AppData\Local\Temp\msohtmlclip1\01\clip_image003.wmz" o:title=""></v:imagedata></v:shape></span>. First, we can prove the existence and uniqueness of the solution of the equation via a prior estimate and Galerkin’s method, then the existence of the family of global attractor is obtained. At last, we can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractor are finite.</span>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
文摘In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations <img src="Edit_e49f9c34-0a5d-4ef2-828f-ba3f0912bed3.png" alt="" />with strong damping terms. We will properly assume the stress term <i>M(s)</i><span style="position:relative;top:6pt;"><v:shape id="_x0000_i1026" type="#_x0000_t75" o:ole="" style="width:27pt;height:17.25pt;"><v:imagedata src="file:///C:\Users\TEST~1.SCI\AppData\Local\Temp\msohtmlclip1\01\clip_image002.wmz" o:title=""></v:imagedata></v:shape></span> and<span style="letter-spacing:-0.2pt;"> nonlinear term g(u<sub>t</sub>)<span style="position:relative;top:6pt;"><v:shape id="_x0000_i1027" type="#_x0000_t75" o:ole="" style="width:27pt;height:17.25pt;"><v:imagedata src="file:///C:\Users\TEST~1.SCI\AppData\Local\Temp\msohtmlclip1\01\clip_image003.wmz" o:title=""></v:imagedata></v:shape></span>. First, we can prove the existence and uniqueness of the solution of the equation via a prior estimate and Galerkin’s method, then the existence of the family of global attractor is obtained. At last, we can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractor are finite.</span>
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.
