Abstract
This paper deals with the existence and uniqueness of solutions of generalized Kirchhoff equations and the family of global attractors for the equation and its dimension estimation. First, the stress term of Kirchhoff equation is properly assumed. When certain conditions are met between the order m and the degree p of Banach space , the existence and uniqueness of the solution of equation are obtained by a prior estimation and Galerkin’s method; Then, the bounded absorption set is obtained by prior estimation, and it is proved that the solution semigroup generated by the equation has a family of global attractors in phase space by using Rellich-Kondrachov compact embedding theorem. Further, the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Fréchet differentiable on ; Finally, the upper bound of Hausdorff dimension and Fractal dimension of is estimated, and the Hausdorff dimension and Fractal dimension are finite.
Highlights
Many scholars have studied the existence of global attractor of Kirchhoff equation with strong dissipative term, [1]-[7] can be referred
In a bounded domain, where m > 1 is a positive integer, p, q, r > 0 is a normal number, if p ≤ r, the existence of global solution will be obtained, if p > max{r, 2q}, for any initial value with negative initial energy, the solution explodes in a finite time
Under the assumption of Kirchhoff stress term, the existence and uniqueness of global solution are obtained by prior estimation and Galerkin’s method
Summary
This paper will study the initial-boundary value problems of the following generalized Kirchhoff equations:. Many scholars have studied the existence of global attractor of Kirchhoff equation with strong dissipative term, [1]-[7] can be referred. In reference [8], scholars considered the following Kirchhoff type wave equation with nonlinear strong damping term ( ) utt − ε1∆ut + α ut p−1 ut + β u q−1 u − φ = ∇u 2 ∆u f ( x),( x,t ) ∈ Ω × R+ , (4). In a bounded domain, where m > 1 is a positive integer, p, q, r > 0 is a normal number, if p ≤ r , the existence of global solution will be obtained, if p > max{r, 2q} , for any initial value with negative initial energy, the solution explodes in a finite time.
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