Resumen | 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 ...
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.
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