Asymptotic Behavior Of The Solutions Of Stochastic Wave Equation

This paper is concerned with the asymptotic behavior of the solution of wave equation.On the one hand,we study the deterministic wave equation with random initial data and prove the existence of weak pullback mean random attractor.On the other hand,we study the existence of a family of semigroups generated by wave equations with multiplicative white noise terms and the global attractors.Actually,the concept of pathwise pullback random attractor has been extensively studied in the literature,but this theory is very restrictive on the diffusion terms of the equations(the diffusion term is generally assumed to be linear).So in 2010,T.Lorenz replace the concept of pathwise random attractor by the mean-square random attractor and weak mean-square random attractor.But this attractor are very restrictive too(suppose the diffusion term satisfies linear growth and Lipschitz conditions).Then in 2019,B.Wang introduce weak pullback mean random attractor.The global attractor is a non-empty compact invariant set that attracts all bounded sets.It is of great significance for the study of infinite-dimensional dynamic systems.Because it contains all possible limit states of the system solution,it can show many information of the system.So,this paper studies the weak pullback mean random attractor and global attractor of wave equation.The first chapter introduces the present status of the theory of attractors,the relevant definitions and inequalities in this paper.The second chapter studies the deterministic wave equation with random initial data,defines a random dynamical system based the solution of the equation and prove the existence of weak pullback mean random attractor.The third chapter studies the asymptotic behavior of the solution of the wave equation with multiplicative white noise.And we fix a sample point,derive a family of semigroups,then we prove the existence of its global attractor.