Abstract
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 g (u) and Kirchhoff stress term M (s) 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 B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.
Highlights
Appropriate assumptions are made for the nonlinear source term g (u ) and Kirchhoff stress term M (s) 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
Abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t ) generated by the equation has a family of the global attractor Ak in the phase space= Ek H 2m+k × H k
The objective of this paper is to study the following initial boundary value problem of the generalized Kirchhoff equation
Summary
The objective of this paper is to study the following initial boundary value problem of the generalized Kirchhoff equation ( ) utt + M. In reference [10], Guoguang Lin, Yunlong Gao studied the existence and uniqueness of global solutions of a class of generalized Kirchhoff-type equations with nonlinear strong damping and their global attractors ( ( )) utt + (−∆)m ut + α + β Dmu 2 (−∆)m= u + g (u) f ( x),( x,t ) ∈ Ω ×[0, +∞). Lin Guoguang and Guan Liping [11] studied the global attractor of a high-order Kirchhoff-type equation with a strong nonlinear damping term and finite dimensional estimation of its Hausdorff dimension and Fractal dimension ( ) utt + M Dmu 2 (−∆)m u + β (−∆)m ut + ∆g (u) = f ( x) where m > 1 , Ω is a bounded domain with smooth homogeneous Dirichlet. Λ 1 is the first eigenvalue of −∆ with homogeneous Dirichlet boundary conditions on Ω
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