Abstract:

Based on the theory of functional diferential equations, theory of semigroup, theory of random dynamical systems and theory of in nite dimensional dynamical systems, this thesis studies the long time behavior of several kinds of in nite dimensional ...
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Based on the theory of functional diferential equations, theory of semigroup, theory of random dynamical systems and theory of in nite dimensional dynamical systems, this thesis studies the long time behavior of several kinds of in nite dimensional dynamical systems associated to partial diferential equations containing some kinds of hereditary characteristics (such as variable delay, distributed delay or memory, etc), including existence and upper semicontinuity of pullback/random attractors and the stability analysis
of stationary (steadystate) solutions. Three important mathematicalphyiscal models are considered, namely, reactiondi usion equation, 2DNavierStokes equation as well as incompressible nonNewtonian uids.
Chapter 1 is devoted to the dynamics of an integer order stochastic reactiondifusion equation with thermal memory when the nonlinear term is subcritical or critical. Notice that our model contains not only memory but also white noise, which means it is not easy to prove the existence and uniqueness of solutions directly. In order to deal with this problem, we need introduce a new variable to transform our model into a system with two equations, and we use the OrnsteinUhlenbeck to transfer this system into a deterministic ones only with random parameter. Then a semigroup method together with the LaxMilgram theorem is applied to prove the existence, uniqueness and continuity
of mild solutions. Next, the dynamics of solutions is analyzed by a priori estimates, and the existence of pullback random attractors is established. Besides, we prove that this pullback random attractors cannot explode, a property known as upper semicontinuity. But the dimension of the random attractor is still unknown. On the other hand, it has been proved that sometimes, especially when selforgnization phenomena, anisotropic di usion, anomalous difusion occurs, a fractional order diferential equation can model this phenomena more precisely than a integer one. Hence, in Chapter 2, we focus on the asymptotical behavior of a fractional stochastic
reactiondifusion equation with memory, which is also called fractional integrodiferential equation. First of all, the OrnsteinUhlenbeck is applied to change the stochastic reactiondifusion equation into a deterministic ones, which makes it more convenient to solve. Then existence and uniqueness of mild solutions is proved by using the LumerPhillips theorem. Next, under appropriate assumptions on the memory kernel and on the magnitude of the
nonlinearity, the existence of random attractor is achieved by obtaining some uniform estimates and solutions decomposition. Moreover, the random attractor is shown to have nite Hausdorf dimension, which means the asymptotic behavior of the system is determined by only a nite number of degrees of freedom, though the random attractor is a subset of an in nitedimensional phase space. But we still wonder whether this random
attractor has inertial manifolds, which means this random attractor needs to be exponentially attracting. Besides, the long time behavior of timefractional reactiondifusion equation and fractional Brownian motion are still unknown.
The rst two chapters consider an important partial function diferential equations with in nite distributed delay. However, partial functional diferential equations include more than only distributed delays; for instance, also variable delay terms can be considered. Therefore, in the next chapter, we consider another signi cant partial functional diferential equation but with variable delay. In Chapter 3, we discuss the stability of stationary solutions to 2D NavierStokes equations when the external force contains unbounded variable delay. Notice that the classic phase space C which is used to deal with diferential equations with in nite delay does not work well for our unbounded variable delay case. Instead, we choose the phase space of continuous bounded functions with limits at1. Then the existence and uni
queness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solutions is established by means of the LaxMilgram theorem and the Schauder xed point theorem. Afterward, the local stability analysis of stationary solutions is carried out by three diferent approaches: the classical Lyapunov function method, the RazumikhinLyapunov technique and by constructing appropriate Lyapunov
functionals. It worths mentioning that the classical Lyapunov function method requires diferentiability of delay term, which in some extent is restrictive. Fortunately, we could utilize RazumikhinLyapunov argument to weak this condition, and only requires continuity of every operators of this equation but allows more general delay. Neverheless, by these methods, the best result we can obtain is the asymptotical stability of stationary solutions by constructing a suitable Lyapunov functionals. Fortunately, we could obtain
polynomial stability of the steadystate in a particular case of unbounded variable delay, namely, the proportional delay. However, the exponential stability of stationary solutions to NavierStokes equation with unbounded variable delay still seems an open problem. We can also wonder about
the stability of stationary solutions to 2D NavierStokes equations with unbounded delay when it is perturbed by random noise. Therefore, in Chapter 4, a stochastic 2D NavierStokes equation with unbounded delay is analyzed in the phase space of continuous bounded functions with limits at1. Because of the perturbation of random noise, the classical Galerkin approximations alone is not enough to prove the existence and uniqueness of weak solutions. By combing a technical lemma and FaedoGalerkin approach, the existence and uniqueness of weak solutions is obtained. Next, the local stability analysis of constant solutions (equilibria) is carried out by exploiting two methods. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. Although it is not possible, in general, to establish the exponential convergence of the stationary solutions, the polynomial convergence towards the stationary solutions, in a particular case of unbounded variable delay can be proved. We would like to point out that the Razumikhin argument cannot be applied to analyze directly the stability of stationary solutions to stochastic equations as we did to deterministic equations. Actually, we need more technical, and this will be our forthcoming paper. We also would like to mention that exponential stability of other
special cases of in nite delay remains as an open problem for both the deterministic and stochastic cases. Especially, we are interested in the pantograph equation, which is a typical but simple unbounded variable delayed diferential equation.We believe that the study of pantograph equation can help us to improve our knowledge about 2D{NavierStokes equations with unbounded delay. Notice that Chapter 3 and Chapter 4 are both concerned with delayed NavierStokes equations, which is a very important Newtonian
uids, and it is extensively applied in physics, chemistry, medicine, etc. However, there are also many important uids, such as blood, polymer solutions, and biological uids, etc, whose motion cannot be modeled pre
cisely by Newtonian uids but by nonNewtonian uids. Hence, in the next two chapters, we are interested in the long time behavior of an incompressible nonNewtonian uids ith delay. In Chapter 5, we study the dynamics of nonautonomous incompressible nonNewtonian uids with nite delay. The existence of global solution is showed by classical Galerkin approximations and the energy method. Actually, we also prove the uniqueness of solutions
as well as the continuous dependence of solutions on the initial value. Then, the existence of pullback attractors for the nonautonomous dynamical system associated to this problem is established under a weaker condition in space C([h; 0];H2) rather than space C([h; 0];L2), and this improves the available results that worked on nonNewtonian uids. However, we still would like to analyze the Hausdor dimension or fractal dimension of the pullback attractor, as well as the existence of inertial manifolds and morsedecomposition. Finally, in Chapter 6, we consider the exponential stability of an incompressible non Newtonian uids with nite delay. The existence and uniqueness of stationary solutions are rst established, and this is not an obvious and straightforward work because of the nonlinearity and the complexity of the term N(u). The exponential stability of steady state solutions is then analyzed by means of four diferent approaches. The rst one
is the classical Lyapunov function method, which requires the diferentiability of the delay term. But this may seem a very restrictive condition. Luckily, we could use a Razumikhin type argument to weaken this condition, but allow for more general types of delay. In fact, we could obtain a better stability result by this technique. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwalllike lemma are also exploited to study the stability, respectively. We would like to emphasize that by using a Gronwalllike lemma, only the measurability of delay term is demanded, but still ensure the exponential stability. Furthermore, we also would like to discuss the dynamics of stochastic nonNewtonian uids with both nite delay and in nite delay. All the problems deserve our attraction, and actually, these are our forthcoming work.
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