Abstract
In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations with strong damping terms. We will properly assume the stress term M(s) and nonlinear term g(ut). First, we can prove the existence and uniqueness of the solution of the equation via a prior estimate and Galerkin’s method, then the existence of the family of global attractor is obtained. At last, we can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractor are finite.
Highlights
This paper intends to study the initial-boundary value problem of higher-orderKirchhoff-type equation ( ) utt + M (−∆)2m u + β (−∆)2m ut + g = f ( x), (1.1) u ( x,=t ) ∂= iu ∂vi 0=, i1, 2, 2m −1, x ∈ ∂Ω, t > 0, (1.2)u= ( x, 0) u0 ( x),u= t ( x, 0) ut ( x), x ∈ Ω ⊂ Rn. (1.3)where m > 1, and m ∈ N +, Ω ∈ Rn (n ≥ 1) is a bounded domain, ∂Ω denotes the boundary of Ω, g is a nonlinear source term, β (−∆)2m ut is a strongly dissipative term, β > 0, f ( x) is an external force term
We can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractors are finite
Lin Guoguang et al studied the existence of global attractors for higher order Kirchhoff type equations with nonlinear strong damping terms in reference [7]: ( ) utt + (−∆)m ut + φ ∇mu 2 (−∆)m u + h = f ( x), u ( x,=t )
Summary
U= ( x, 0) u0 ( x),u= t ( x, 0) ut ( x), x ∈ Ω ⊂ Rn. where m > 1 , and m ∈ N + , Ω ∈ Rn (n ≥ 1) is a bounded domain, ∂Ω denotes the boundary of Ω , g (ut ) is a nonlinear source term, β (−∆)2m ut is a strongly dissipative term, β > 0 , f ( x) is an external force term. There have been many achievements in the study of the long-term behavior of the solution of Kirchhoff-type equation, for details, refer to references ([1] [2] [3] [4] [5]). Lin Guoguang et al studied the existence of global attractors for higher order Kirchhoff type equations with nonlinear strong damping terms in reference [7]:. 1, 2, , m −1, x ∈ ∂Ω, t > 0, u (= x, 0) u= 0 , ut u1 ( x), x ∈ Ω They proved the existence and uniqueness of the solution of the equation by using prior estimation and Galerkin’s method, and obtained that the attractor exists in space H 2m (Ω)× H m (Ω). Guoguang Lin and Changqing Zhu studied asymptotic state of solutions for a class of nonlinear higher order Kirchhoff type equations in reference [8]:. Please refer to references ([9]-[15])
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