JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (12): 1-11.doi: 10.6040/j.issn.1671-9352.0.2018.709

   

Attractor family and dimension for a class of high-order nonlinear Kirchhoff equations

LIN Guo-guang, LI Zhuo-xi   

  1. School of Mathematics and Statistics, Yunnan University, Kunming 650500, Yunnan, China
  • Published:2019-12-11

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Abstract: The initial boundary value problem for a class of high-order Kirchhoff equations with nonlinear nonlocal source terms and strong damping terms is studied. For the nonlinear nonlocal source term and the Kirchhoff stress term, the existence and uniqueness of the global solution of the equation are firstly proved by Galerkin finite element method and a prior estimate. Then the bounded absorption set is obtained by a prior estimate, so the global attractor family of high-order nonlinear Kirchhoff equation is obtained. By linearizing the equation and proving the Frechet differentiable of the solution semigroup, it further proves the decay of the volume element of the linearization problem. Finally, the Hausdorff dimension and Fractal dimension of the global attractor family are proved to be finite.

Key words: high-order Kirchhoff equation, Galerkin finite element method, global attractor family, Hausdorff dimension, Fractal dimension

CLC Number: 

  • O175.29
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