Long-time dynamics of non-autonomous random Boissonade equations driven by nonlinear colored noises
DOI:
https://doi.org/10.12775/TMNA.2025.061Keywords
Random Boissonade equations, nonlinear colored noise, nonuniqueness, multivalued non-autonomous random dynamical system, long-time dynamics, asymptotic autonomyAbstract
Consider non-autonomous random Boissonade equations on a three-dimensional bounded domain with boundary of Lipschitz type, where homogeneous boundary conditions of Dirichlet type are equipped. This type of equations is a activator-inhibitor model that exhibits Turing structures for describing the relations between the genuine homogeneous 2D systems and the 3D monolayers. The underlying feature of the problem under consideration is a non-Lipschitz condition imposed on the nonlinear function controlling intensity of colored noise, leading to possible nonuniqueness of solutions with given initial data. A class of multivalued semiflows having non-autonomy and randomness is defined. The measurability and continuity properties of such semiflows are studied. Moreover, the long-time dynamics in the pullback sense for the semiflows is studied using the idea of energy equation. We also investigate the asymptotically autonomous robustness for the equations on a three-dimensional bounded domain with smooth boundary.References
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