Abstract
In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: ${u_{tt}} + {( - \Delta )^m}{u_t} + {\left( {\alpha + \beta\left\| {{\nabla ^m}u} \right\|^2} \right)^{q}}{( - \Delta )^m}u + g(u) = f(x)$. At first, we do priori estimation for the equations to obtain two lemmas and prove the existence and uniqueness of the solution by the lemmas and the Galerkin method. Then, we obtain to the existence of the global attractor in $H_0^m(\Omega ) \times {L^2}(\Omega )$ according to some of the attractor theorem. In this case, we consider that the estimation of the upper bounds of Hausdorff for the global attractors are obtained. At last, we also establish the existence of a fractal exponential attractor with the non-supercritical and critical cases.
Highlights
In this paper, we are concerned with the existence of global attractor for the following nonlinear Higher-order Kirchhofftype equations: utt + (−∆)mut ( α+β ∥∇mu∥2)q(−∆)mu + g(u) = f (x), (x, t) ∈ Ω × [0, +∞), (1.1)u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, (1.2)
We obtain to the existence of the global attractor in H0m(Ω) × L2(Ω) according to some of the attractor theorem
We consider that the estimation of the upper bounds of Hausdorff for the global attractors are obtained
Summary
We are concerned with the existence of global attractor for the following nonlinear Higher-order Kirchhofftype equations: utt. Where Ω is a bounded domain in RN with the smooth boundary ∂Ω, σ(s), φ(s) and f (s) are nonlinear functions, and h(x) is an external force term They prove that in strictly positive stiffness factors and supercritical nonlinearity case, there exists a global finite-dimensional attractor in the natural energy space endowed with strong topology. Under these assumptions, we prove the existence and uniqueness of solution, we obtain the global attractors for the problems (1.1)-(1.3). J. & et al, 2014; Zhijian Yang & Zhiming Liu, 2015; Igor Chueshov, 2012)
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