Continuous Wong–Zakai Approximations of Random Attractors for Quasi-linear Equations with Nonlinear Noise

• Autores: Yangrong Li, Shuang Yang, Qiangheng Zhang
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 3, 2020
• Idioma: inglés
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• Resumen

  • We consider a family of random quasi-linear equations driven by nonlinear Wong–Zakai noise and parameterized by the non-zero size λ of noise. After proving the existence of a random attractor Aλ(ω) in the square Lebesgue space, we then show that there is a residual dense subset of the space of nonzero real numbers such that, under the Hausdorff metric, the map λ→Aλ(θsω) is continuous at all points of the residual dense set, where θs is a group of self-transformations on the probability space. We also prove that as λ→±∞ the random attractor converges upper-semicontinuously to the global attractor of the deterministic quasi-linear equation. The upper semi-continuity result is new for nonlinear noise, while, the lower semi-continuity result is new even for linear noise. The theory of Baire category is the main tool used to prove the residual continuity.