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On the regularity of global attractors  [PDF]
Monica Conti,Vittorino Pata
Mathematics , 2009,
Abstract: This note is focused on a novel technique in order to establish the boundedness in more regular spaces for global attractors of dissipative dynamical systems, without appealing to uniform-in-time estimates. As an application of the abstract result, the semigroup generated by the strongly damped wave equation $$u_{tt}-\Delta u_t-\Delta u+\phi(u)=f$$ with critical nonlinearity is considered, whose attractor is shown to possess the optimal regularity.
The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term  [PDF]
Yuting Sun, Yunlong Gao, Guoguang Lin
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2016, DOI: 10.4236/ijmnta.2016.54019
Abstract: We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: \"\". For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in \"\" proved by Galerkin method, and deal with the global attractors.
Counterexamples to the regularity of Mane projections and global attractors  [PDF]
Alp Eden,Varga Kalanarov,Sergey Zelik
Mathematics , 2011,
Abstract: We study the global attractors of abstract semilinear parabolic equations and their projections to finite-dimensional planes. It is well-known that the attractor can be embedded into the finite-dimensional inertial manifold if the so-called spectral gap condition is satisfied. We show that in the case when the spectral gap condition is violated, it is possible to construct the nonlinearity in such way that the corresponding attractor cannot be embedded into any finite-dimensional Log-Lipschitz manifold and, therefore, does not possess any Mane projections with Log-Lipschitz inverse. In addition, we give an example of finitely smooth nonlinearity such that the attractor has finite Hausdorff but infinite fractal dimension.
Global Attractors for a Class of Generalized Nonlinear Kirchhoff-Sine-Gordon Equation  [PDF]
Ruijin Lou, Penghui Lv, Guoguang Lin
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2016, DOI: 10.4236/ijmnta.2016.51008
Abstract: In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation?\"\". By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.
Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy
Gang Li,Linghui Hong,Wenjun Liu
Journal of Function Spaces and Applications , 2012, DOI: 10.1155/2012/530861
Abstract: We consider viscoelastic wave equations of the Kirchhoff type ?(‖?‖22∫)Δ
The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping  [PDF]
Yunlong Gao, Yuting Sun, Guoguang Lin
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2016, DOI: 10.4236/ijmnta.2016.54018
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: \"\". At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.
Smooth attractors for the quintic wave equations with fractional damping  [PDF]
Anton Savostianov,Sergey Zelik
Mathematics , 2013,
Abstract: Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors of finite Hausdorff and fractal dimension is verified.
The Global Attractors for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Linear Damping and Source Terms  [PDF]
Liang Guo, Zhaoqin Yuan, Guoguang Lin
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2015, DOI: 10.4236/ijmnta.2015.42010
Abstract: In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a nonlinear viscoelastic wave equation with strong damping, linear damping and source terms. Then we study the global attractors of the equation.
The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays*  [PDF]
Guoguang Lin, Fangfang Xia, Guigui Xu
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2013, DOI: 10.4236/ijmnta.2013.24029
Abstract:

In this paper, we study the global and pullback attractors for a strongly damped wave equation with delays when the force term belongs to different space. The results following from the solution generate a compact set.

Global Low Regularity Solutions of Quasi-linear Wave Equations  [PDF]
Yi Zhou,Zhen Lei
Mathematics , 2007,
Abstract: In this paper we prove the global existence and uniqueness of the low regularity solutions to the Cauchy problem of quasi-linear wave equations with radial symmetric initial data in three space dimensions. The results are based on the end-point Strichartz estimate together with the characteristic method.
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