Liu Ruonan, Caraballo Tomás
School of Mathematics and Statistics, Xuzhou University of Technology, Jiangsu 221008, People's Republic of China.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain.
Chaos. 2024 Oct 1;34(10). doi: 10.1063/5.0223756.
In this paper, we will show two approaches to analyze the dynamics of a stochastic partial differential equation (PDE) with long time memory, which does not generate a random dynamical system and, consequently, the general theory of random attractors is not applicable. On the one hand, we first approximate the stochastic PDEs by a random one via replacing the white noise by a colored one. The resulting random equation does generate a random dynamical system which possesses a random attractor depending on the covariance parameter of the colored noise. On the other hand, we define a mean random dynamical system via the solution operator and prove the existence and uniqueness of weak pullback mean random attractors when the problem is driven by a more general white noise.
