Abstract
In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.
Highlights
In this paper, we considers the initial-boundary value problem of the following higher-order nonlinear Kirchhoff-type equations utt + a(t)(−∆)m ut ∆2mu N( Dmu(t) q )(−∆)m u q = I ( x) (1)u(x, t) = 0, ∂iu = 0, i = 1, 2,⋅ ⋅ ⋅, 2m −1, x ∈ ∂Ω, t > 0, (2) ∂vi u(x, 0) = u0 (x), ut (x, 0) = u1(x), x ∈ Ω ⊂ Rn. (3)
Studied the well posedness and long-time behavior of the solution of the initial boundary value problem of the equation utt −σ ( Du 2 )∆ut −φ( Du 2 )∆u + g(u) = h(x), By assuming the Kirchhoff term, he proved the existence and uniqueness of weak solution and the existence of a finite dimensional global attractor in the natural energy space with partial strong topology, and further proved that the attractor is strong under non supercritical conditions
Theorem 2 According to lemma 1 and theorem 1, the initial boundary value problem (1) - (3) has a family of global attractors
Summary
We considers the initial-boundary value problem of the following higher-order nonlinear Kirchhoff-type equations utt. Studied the well posedness and long-time behavior of the solution of the initial boundary value problem of the equation utt −σ ( Du 2 )∆ut −φ( Du 2 )∆u + g(u) = h(x) , By assuming the Kirchhoff term, he proved the existence and uniqueness of weak solution and the existence of a finite dimensional global attractor in the natural energy space with partial strong topology, and further proved that the attractor is strong under non supercritical conditions. On the basis of chueshov Igor [9], Guoguang Lin [10] and others studied the long-term behavior of the initial boundary value problem for a class of nonlinear strongly damped higher order Kirchhoff type equation with m = 1, q = 2, a(t) = 1 and. We overcome this problem successfully and get more extensive research methods and theoretical results
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