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Invariant measure and large time dynamics of the cubic Klein-Gordon equation in \(3D\). (English) Zbl 1457.60110

Summary: In this paper we construct an invariant probability measure concentrated on \(H^2(K)\times H^1(K)\) for a general cubic Klein-Gordon equation (including the case of the wave equation). Here \(K\) represents both the 3-dimensional torus or a bounded domain with smooth boundary in \({\mathbb{R}}^3\). That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic sense. We also establish qualitative properties of the constructed measure. This work extends the fluctuation-dissipation-limit approach to PDEs having only one (coercive) conservation law.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
28D05 Measure-preserving transformations
35B40 Asymptotic behavior of solutions to PDEs
35L71 Second-order semilinear hyperbolic equations
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