| Fractional differential equations have wide applications as well as developments in physics,chemistry,biology,finance,engineering and other fields of science.Cur-rently,the dynamical behavior of fractional partial differential equations driven by linear noise has been investigated.This thesis is devoted to the random dynamics by means of pullback random attractors,weak pullback mean random attractors and invariant measures for several classes of fractional partial differential equations driv-en by nonlinear white(colored)noise.The structure,content,difficulty as well as innovation of the thesis are illustrated as follows.The first part of the thesis is concerned with the well-posedness and random dy-namics of the fractional nonclassical diffusion equations defined on unbounded do-mains driven by nonlinear colored noise.Precisely,the well-posedness as well as the existence of energy equations of the equations are proved in the fractional Sobolev s-pace Hs(RN)for all s?(0,1].The idea of energy method due to J.M.Ball is used to prove that the equation has a unique pullback random attractor in Hs(RN)when the nonlinear drift and diffusion terms have arbitrary and superlinear growth rates,respectively in order to overcome the difficulty of the non-compactness of Sobolev embeddings on unbounded domains.For the case of addictive colored noise,we prove that those attractors converge to the random attractors of the equations with addictive white noise.This is the first time to study attractors of fractional nonclassical diffusion equations,and the results are even new when s=1.The second part of the thesis is devoted to the random dynamics of a class of weakly dissipative fractional wave equations defined on unbounded domains perturbed by nonlinear colored noise.It is difficult to derive uniform estimates of the solution-s since the equations contains many fractional Laplacian terms and nonlinear terms.In order to overcome the non-compactness of Sobolev embeddings on unbounded do-mains,we employ the methods of uniform tails-estimates and spectral decomposition to prove that the equation has a unique pullback random attractor in the Sobolev space Hs(RN)× Hs(RN)for all s?(0,1)when the nonlinear drift and diffusion terms have subcritical and superlinear growth rates,respectively.For the case of colored noise,the upper semi-continuity of those attractors is also establishedThe third part of the thesis deals with the well-posedness as well as existence of weak pullback mean random attractors of the fractional FitzHugh-Nagumo systems on unbounded domains driven by nonlinear white noise.The main difficulty of proving the well-posedness of the systems is how to treat the locally Lipschitz white noise.We use approximate methods to prove the well-posedness of the systems in the case of regular addictive noise,general addictive noise,globally Lipschitz noise and locally Lipschitz noise.Based upon the well-posedness of the systems,we define a mean random dynamical system,and prove that this systems has a unique weak pullback mean random attractor in a Bochner spaceThe last part of the thesis focuses on the existence of invariant measures of the fractional FitzHugh-Nagumo systems defined on unbounded domains driven by a fam-ily of nonlinear white noise.It is difficulty to obtain the tightness of probability dis-tributions of the solutions due to the non-compactness of Sobolev embeddings on un-bounded domains and the lack of regularity of one component of solutions.We em-ploy the approach of uniform tail-estimates to prove the uniform smallness of the mean square solutions outsider a sufficiently large bounded ball.When the initial data have regularity,we also prove that the mean square solutions have corresponding regularity Based upon those uniform estimates,we prove the tightness of probability distributions of the solutions,and hence the existence of invariant measuresWe remark that the results of the thesis are new when the fractional Laplace op-erator is replaced by the standard Laplace operator. |
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