Archive
Special Issues Volume 7, Issue 1, February 2019, Page: 21-29
Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation
Guoguang Lin, School of Mathematics and Statistics, Yunnan University, Kunming, China
Ying Jin, School of Mathematics and Statistics, Yunnan University, Kunming, China
Received: Apr. 1, 2019;       Accepted: May 10, 2019;       Published: Jun. 3, 2019
Abstract
In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.
Keywords
Kirchhoff Equation, The Existence and Uniqueness of Solutions, Global Attractor Family, Hausdorff Dimension, Fractal Dimension
Guoguang Lin, Ying Jin, Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation, American Journal of Applied Mathematics. Vol. 7, No. 1, 2019, pp. 21-29. doi: 10.11648/j.ajam.20190701.14
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