The Global and Exponential Attractors for the Higher-order Kirchhoff-type Equation with Strong Linear Damping

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.