The Global and Exponential Attractors for the Higher-order \\[6pt] Kirchhoff-type Equation with Strong Linear Damping

Guoguang Lin and Yunlong Gao

Journal of Mathematics Research, 2017, vol. 9, issue 4, 145-167

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.

Keywords: Nonlinear Higher-order Kirchhoff type equation; Galerkin method; The existence and uniqueness; The Global attractor; Huasdorff dimensions; The Exponential attractor (search for similar items in EconPapers)
Date: 2017
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