Stability Of Attractors For Stochastic Delay Partial Differential Equations

In this thesis,we study the time-dependent stability of pullback random attractors(PRAs).The abstract results of this kind of stability are established and applied to sev-eral kinds of stochastic delay partial differential equations,including Swift-Hohenberg equation,Brinkman-Forchheimer equation,Navier-Stokes equation and p-Laplacian equation.In Chapter 1,we introduce the research background,significance and current sit-uation of this paper as well as the main results of this paper.In Chapter 2,we establish three abstract results about the stability of PRAs:(1)The longtime stability of the PRA ?={?(?,?)?.More precisely,there exists a nonempty compact set such that the PRA upper semi-converges to this set under the sense of Hausdorff semi-distance as the time parameter ? tends to minus infinity,and this set is called the backward controller if it is the smallest.(2)The regular asymptot-ically backward autonomy of PRAs,that is,we prove the PRA upper semi-converges to its corresponding(autonomous)random attractor in a highly regular target space when the initial value space is low in regularity and ? goes to minus infinity.(3)The asymptotically forward autonomy of bi-spatial PRAs.In addition,the bi-spatial PRA converges to bi-spatial random attractor as ? tends to positive infinity if the bi-spatial random attractor corresponding to the bi-spatial PRA is a singleton or the low-limit set of the bi-spatial PRA.In Chapter 3,we consider the stochastic modified Swift-Hohenberg equations with abstract delay and non-autonomous forcing,where the delay term is pointwise Lipschtz continuous as well as the non-autonomous force term is backward limitable,and without restricting the upper bound of Lipschitz coefficients.We apply the first result to discuss the longtime stable dynamics of this equation under the above weak assumptions.Furthermore,we provide two examples to show that the assump-tions of the delay term is reasonable at the end of this chapter.In Chapter 4,we study the regular asymptotically backward autonomous dynam-ics of stochastic Brinkman-Forchheimer equations with general delay term.Due to the Wiener process is almost everywhere non-differentiable,we use the spectral decompo-sition technique and Ascoli-Arzela theorem to prove the backward pullback asymptot-ic compactness of solutions in the regular space,and then obtain a regular backward compact pullback random attractor ?.We then verify the assumptions of the second result to show that the regular asymptotically backward autonomy of A.In Chapter 5,we consider the stochastic Navier-Stokes equations with general autonomous delay term,and apply the third abstract result to this equation.Because of the solution of this equation has no higher regularity,we prove the forward flattening of solutions and apply the Ascoli-Arzela theorem to obtain the bi-spatial forward pullback asymptotic compactness of solutions,and then establish the existence and uniqueness of the bi-spatial forward compact pullback random attractor ?.We show that ?has a forward controller and its asymptotically forward autonomy by the convergence of solutions from non-autonomy to autonomy in the initial space and the recurrence of absorbing sets.In the last chapter,we study the random stable dynamics of p-Laplacian equations with multiplicative noise and variable delay on unbounded domains.For this pur-pose,we need to solve the problem of the non-compactness of Sobolev embeddings on unbounded domains and the problem of the measurability of backward compact non-autonomous attractors.For the above problems,we use the backward uniform tail estimates of solutions as well as prove the attractor is same in two different domains of attraction,and then obtain a longtime stable backward compact PRA.Finally,we also prove the zero-memory stability of this attractor.