The Evolution of Evolutionary Equations

Sell, George R.; You, Yuncheng

doi:10.1007/978-1-4757-5037-9_1

Cited by 30 publications

(43 citation statements)

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“…We obtain a separable Hilbert space (D(A r ), ·, · D(A r ) , · D(A r ) ) by setting ·, · D(A r ) := A r ·, A r · . For more details for the set up above, see for example [12,21,25,29]. Let (Ω, F , P) be a probability space equipped with a filtration satisfying the usual conditions.…”

Section: Settingmentioning

confidence: 99%

Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

Kazashi¹

2018

Electron. Commun. Probab.

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An implicit Euler-Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal L 2 -regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.

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Section: Settingmentioning

confidence: 99%

Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

Kazashi¹

2018

Electron. Commun. Probab.

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“…It is well known that the regularity and finite dimensionality of the attractor play an important role in many interesting fields for applications, and there exist a number of literatures on this topic, such as [3][4][5][6][7][8][9], and references therein.…”

Section: Theoremmentioning

confidence: 99%

Regularity and finite dimensionality of attractor for plate equation on R n

Xiao

2010

Appl. Math. Mech.-Engl. Ed.

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“…More complex structures of attractors that include multiple equilibria and other orbits put this problem into a different perspective. As recognized recently in [1] page 353 "globally dissipative dynamics of nonlinear wave equation or hyperbolic evolutionary equations with boundary damping remains an open problem". The presence of boundary dissipation is no longer represented by a bounded (on the phase space) operator, in contrast to the interior (full or localized) damping [H-R,F1,F2,F3].…”

Section: • (Iii) Fractal Dimension Of Attractorsmentioning

confidence: 99%

Attractors and their structure for semilinear wave equations with nonlinear boundary dissipation

2009

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Long time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a finite dimensional attractor. It is known [CEL1] that the attractor consists of all full trajectories emanating from the set of stationary points. Under the additional assumption that the set of stationary points is finite it is proved that every solution converges to some stationary points at an exponential rate.

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The Evolution of Evolutionary Equations

Cited by 30 publications

References 0 publications

Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

Regularity and finite dimensionality of attractor for plate equation on R n

Attractors and their structure for semilinear wave equations with nonlinear boundary dissipation

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