Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations  

作　　者：Guoguang Lin Lujiao Yang Guoguang Lin;Lujiao Yang(Department of Mathematics, Yunnan University, Kunming, China)

机构地区：[1]Department of Mathematics, Yunnan University, Kunming, China

出　　处：《Advances in Pure Mathematics》2021年第4期317-333,共17页理论数学进展（英文）

摘　　要：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>_0<em>k</em> 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>_<em>k</em></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_k</em>. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor <em>A_k</em> was obtained.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>_0<em>k</em> 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>_<em>k</em></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_k</em>. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor <em>A_k</em> was obtained.

关 键 词：Generalized Kirchhoff Equation The Existence and Uniqueness of Solution A Family of the Global Attractor Dimension Estimation 

分 类 号：O17[理学—数学]