In this paper, we study the inertial manifolds for a class of the Kirchhoff-type
equations with strongly damped terms and source terms. The inertial manifold
is a finite dimensional invariant smooth manifold that contains the global
attractor, attracting the solution orbits by the exponential rate. Under appropriate
assumptions, we firstly exert the Hadamard’s graph transformation
method to structure a graph norm of a Lipschitz continuous function, and
then we prove the existence of the inertial manifold by showing that the spectral
gap condition is true.

本文主要研究了强阻尼波动方程的整体吸引子由光滑流形来逼近。构造了强阻尼波动方程的一个非线性近似惯性流形，并得到了该近似惯性流形逼近整体吸引子的阶数估计。
In this paper, the global attractor approximation by smooth manifold is considered in strongly damped equation. A nonlinear approximate inertial manifold of strongly damped wave equation is constructed. The order of approximation of the inertial manifold to the global attractor is obtained．

Abstract:
This paper considers the long-time behavior for a system of coupled wave equations of higher-order Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of the inertial manifold while such equations satisfy the spectral interval condition.

Abstract:
In this paper, we give a new approximate inertial manifold and application to nonlinear elliptic boundary value problems. The approximate solution possesses over double convergence rate compared with the standard Galerkin approximate solution. And an example is given. The result of the numerical simulates show that the Post-Galerkin Method is very effective in improving precision of the ap- proximate solution.

Abstract:
In this paper, stochastic inertial manifold for damped wave equations subjected to additive white noise is constructed by the Lyapunov-Perron method. It is proved that when the intensity of noise tends to zero the stochastic inertial manifold converges to its deterministic counterpart almost surely.

Abstract:
We consider inertial manifolds and their approximation for a class of partial differential equations with a nonlocal Laplacian operator $-(-\Delta)^{\frac{\alpha}{2}}$, with $0<\alpha<2$. The nonlocal or fractional Laplacian operator represents an anomalous diffusion effect. We first establish the existence of an inertial manifold and highlight the influence of the parameter $\alpha$. Then we approximate the inertial manifold when a small normal diffusion $\varepsilon \Delta$ (with $\varepsilon \in (0, 1)$) enters the system, and obtain the estimate for the Hausdorff semi-distance between the inertial manifolds with and without normal diffusion.

Abstract:
The concept of approximate inertial manifold is related to the study of the long time behavior of dissiputive partial differential equations. In the present paper, the authors construct two approximate inertial manifolds for the nonlinear Sobolev Galpern equations. The authors show that the non flat approximate inertial manifold Σ and the flat approximate inertial manifold Σ_0=P_mH have the same order of approximation to the global attractor.

Abstract:
This paper studies theexponential attractor for a class of the Kirchhoff-type equations
with strongly damped terms and source terms. The exponential attractor
is also called the inertial fractal set, which is
an intermediate step between global attractors and inertial manifolds.Obtaining a set that
attracts all the trajectories of the dynamical system at an exponential rate by
the methods of Eden A. Under appropriate assumptions, we firstly construct an
invariantly compact set. Secondly, showing the solution semigroups of the Kirchhoff-type
equations is squeezing and Lipschitz continuous. Finally, the finite fractal
dimension of the exponential attractor is obtained.

Abstract:
By applying Rohlin's result on the classification of homomorphisms of Lebesgue space, the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular perturbation parameter. This approximation can be seen as the Smolukowski--Kramers approximation as time goes to infinity. However, as time goes infinity, the approximation changes with the small parameter, which is different from the approximation on a finite interval.

Abstract:
For two particular choices of the three parameters in the FitzHugh-Nagumo model the equilibrium points are found. The corresponding phase portrait around them is graphically represented allowing us to delimit an absorbing domain. Then the Jolly-Rosa-Temam numerical method is applied in order to study the approximate inertial manifold for the model. To this aim the own numerical code of the first author is used.