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The Inertial Manifold for Class Kirchhoff-Type Equations with Strongly Damped Terms and Source Terms  [PDF]
Guoguang Lin, Xiangshuang Xia
Applied Mathematics (AM) , 2018, DOI: 10.4236/am.2018.96050
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.
Approximate Inertial Manifold of Strongly Damped Wave Equation

张素方, 张建文
Pure Mathematics (PM) , 2015, DOI: 10.12677/PM.2015.56040
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.
The Inertial Manifold for a Class of Nonlinear Higher-Order Coupled Kirchhoff Equations with Strong Linear Damping  [PDF]
Guoguang Lin, Lingjuan Hu
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2018, DOI: 10.4236/ijmnta.2018.72003
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.

Huang Aixiang Liu Zhixing Zhang Yinti,

计算数学 , 2001,
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.
Stochastic inertial manifolds for damped wave equations  [PDF]
Zhenxin Liu
Mathematics , 2006,
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.
Approximation of the inertial manifold for a nonlocal dynamical system  [PDF]
Xingjie Yan,Jinchun He,Jinqiao Duan
Mathematics , 2014,
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.
Approximate Inertial Manifolds for the Nonlinear Sobolev Galpern Equations

SHANG Y-Dong,GUO Bo-Ling,

数学物理学报(A辑) , 2004,
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.
The Exponential Attractor for a Class of Kirchhoff-Type Equations with Strongly Damped Terms and Source Terms  [PDF]
Guoguang Lin, Xiangshuang Xia
Journal of Applied Mathematics and Physics (JAMP) , 2018, DOI: 10.4236/jamp.2018.67125
Abstract: This paper studies the exponential 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.
Approximation of the random inertial manifold of singularly perturbed stochastic wave equations  [PDF]
Yan Lv,Wei Wang,Anthony Roberts
Mathematics , 2012,
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.
Further results on approximate inertial manifolds for the FitzHugh-Nagumo model  [cached]
Simona Cristina Nartea,Adelina Georgescu
Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali , 2009,
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.
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