The Non-autonomous Power System Pulls Back The Topological Properties Of The Attractor

This thesis studies the long time behavior of pullback attractors for non-autonomous dynamical systems.First,we establish a unified theoretical criterion on the existence and upper semi-continuity of bi-spatial pullback attractors for non-autonomous cocycles.That is,when a family of non-autonomous cocycles is convergent,uniformly pullback absorbing in the initial space,and it is uni-formly pullback asymptotically compact in both initial and non-initial spaces,we obtain such a unified result.As an application,we consider the following non-autonomous and stochastic FitzHugh-Nagumo equation on Rn,n?N,(?)where ?,?>O,f is the nonlinearity,g1,g2 are body forces and W1,W2 are random noises.By using some new Gronwall-type inequalities and techniques of positive and negative truncations,we prove that such a coupling equation has a bi-spatial pullback attractor when the initial space is L2(Rn)2 and the non-initial space is H1(Rn)× L2(Rn).Then,we give a new theoretical framework to discuss the bound of Lq-box-counting dimensions of random attractors for stochastic partial differential equa-tions on an unbounded domain.In particular,we study the following stochastic degenerate parabolic equation with multiplicative noise on RN,N? 2,(?)where ?>0,?;? R is the density of noise,W is a two-sided real-valued Wiener process on a probability space(?,F,P),g is the external force,a is the diffusion coefficient,and f is the nonlinearity.Under some weak assumptions for the force and the nonlinearity,we prove the existence of a unique(L2,D01,2(?)Lq)-random attractor for any q?[2,(p-2)I+2],where p—1 is the order of the nonlinearity and I is a given integer such that the force is(I+1)-times integrable.On the other hand,by using truncation and splitting techniques,and also induction methods,we prove that a priori estimate is uniform with respect to the density of noise,which leads to the upper semi-continuity result of the obtained attractors as the density tends to a constant(including zero)under the topology of the non-initial space.Moreover,we show that the Lq-box-counting dimension of the obtained attractors is bounded.Finally,we establish some new abstract criteria on the backwards topological property of pullback attractors for an evolution process.We prove that an evolu-tion process has a backwards compact attractor,i.e.the union of attractors over the past time is pre-compact,if it has an increasing,bounded and pullback ab-sorbing set,and it is backwards pullback limit-set compact(or equivalently back-wards pullback asymptotically compact or backwards pullback flattening).We apply these abstract criteria and consider the following non-autonomous damped 3D Navier-Stokes equation on a smooth bounded domain(?)(?)R3:where ??R,?>0 is the kinematic viscosity,?>0 and ??1 are constants in the nonlinear damping,u and p denote the velocity field and pressure field,g is a non-autonomous force.By applying the Gagliardo-Nirenberg inequality and spectrum decomposition,we obtain a pullback attractor in a square integrable space if the order of the damping is larger than three,and further in a Sobolev space if the order belongs to(3,5).The latter of which improves the best range[7/2,5)given in literatures so far.We use some new hypotheses on the force here,which are weaker than those given in literatures.More importantly,we prove that the obtained attractor is backwards compact in the corresponding space.