Dynamics of deterministic and stochastic systems containing colored noise and delay or memory

Abstract

This thesis is concerned with the dynamics of deterministic systems with memory, as well as stochastic systems with nonlinear white (colored) noise, with or without delay. In the three parts of the thesis, we specifically study nonlocal semilinear degenerate heat equations in Chapter 1, random p-Laplace equations in Chapter 2-Chapter 3, and stochastic modified Swift-Hohenberg lattice systems in Chapter 4-Chapter 5. In Chapter 1, we consider the existence of global attractors of nonlocal semilinear degenerate heat equations with degenerate memory on a bounded domain. We use the Faedo-Galerkin method to prove the existence of solutions of the non-degenerate heat equation, and then use it to approximate the equation obtained by the Dafermos transform, which is subsequently combined with the properties of the defined associated operators to obtain the existence, uniqueness, and regularity of solutions to the original degenerate problem. In addition, we establish an autonomous dynamical system and thereafter show the existence of the global attractor of the original problem. In Chapter 2, we show the continuity of pullback random attractors for random p-Laplace equations driven by nonlinear colored noise on unbounded domains. We establish abstract results for the existence and residual dense continuity of a unique pullback random bi-spatial attractor. In its proof, we may consider the larger metric space of all closed bounded sets in regular space and will use the abstract Baire residual theorem and the Baire density theorem. After that, we justify the existence and the residual dense continuity of pullback random bi-spatial attractors of the random p-Laplace equation on both initial space and regular space. In Chapter 3, we prove the existence of pullback random attractors for random p-Laplace equations with nonlinear color noise and infinite delays on a bounded domain. To obtain the existence of weak solutions to the equation, we use the traditional Galerkin approximation technique. Since the data for the problem do not satisfy a Lipschitz continuity condition, the weak solution may not be unique. By proving the continuity and the cocycle property of solutions, as well as the measurability of setvalued maps generated by multiple solutions, thus generate a multi-valued random dynamical system. As a further result, we prove the existence and measurability of a pullback attractor in the framework of multi-valued random dynamical systems. In Chapter 4, we take into account the existence and the limiting behavior of periodic measures for the stochastic delay modified Swift-Hohenberg lattice systems with nonlinear white noise. We need to prove the tightness of distribution laws of solutions to the system, and then combine it with Krylov- Bogolyubov’s method to prove the existence of periodic measures of the lattice system. Then, by strengthening the as- sumptions, we prove that the set of all periodic measures is weakly compact, and also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero. In Chapter 5, we investigate the asymptotic stability of evolution systems of probability measures for stochastic discrete modified Swift-Hohenberg equations with nonlinear white noise. We use the technique of cut-off functions and a stopping time to establish the well-posedness of the system in the Bochner space. Based on uniform tail estimates, we can show the tightness of distribution laws of solutions and thus the existence of evolution systems of probability measures. Moreover, we discuss that the evolution system of probability measures of the limit equation is the limit of the evolution system of probability measures when the noise intensity tends to a certain value.