Abstract

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 E0 and Ek, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set B0k on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup S(t) of the equation has the family of the global attractor Ak in space Ek. Finally, we prove that the solution semigroup S(t) is Frechet differentiable on Ek via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor Ak.

Highlights

  • 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 E0 and Ek, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method

  • Kirchhoff equation is a kind of important nonlinear wave equation, which is widely used in engineering physics, especially provides a strong support for measuring bridge vibration

  • Kirchhoff equation originates from a physical model, which is obtained by German physicist Gustav Robert Kirchhoff [1] when he studied the transverse vibration of elastic string: ρh

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Summary

Introduction

We mainly study the initial boundary value problem of the Kirchhoff equation with high order strong damping:. Igor Chueshov studied the Kirchhoff equation with strong dissipative ( ) term for =m 1,=p 2,=β σ ∇u 2 in reference [3]: ( ) ( ) utt −σ ∇u 2 ∆ut −φ ∇u 2 ∆u + g (u) = f ( x), They proved that its weak solution exists and is unique. At the same time, summing up previous experience, we discuss the difficult problem of the relationship between order m and p in the rigid term and ρ in the nonlinear term, and get some theoretical results about the long time behavior of the equation

The Basic Assumptions
Existence and Uniqueness of Solutions
Global Attractor and Dimension Estimation

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