The Existence Of Random Attractors For A Stochastic Damped Wave Equation With Multiplicative Noise On Unbounded Domains

Stochastic damped wave equation has rich actual background. The existence andasymptotical behavior of its solutions and the existence and regularity of its globalattractor have become the most hot topics. The deterministic damped wave equationhas been studied well. In the real world, There are uncertain factors, and manyproblems evolve with the interference of uncertainty. This leads to the fact thatrandom model must be considered. Recently, researches about stochastic dampedwave equation especially receive more and more attentions. Many scholars studystochastic damped wave equation,and have got some beautiful results.In this paper, we study the random attractor for a kind of stochastic damped waveequation with non-linear multiplicative noise in unbounded domains. The key workwe do are dealing with random item and overcoming the lacking of the compactnessof Sobolev embedding. Fist, by the use of O-U process, the equation is transformedinto a diferential equation with random coefcients. On the basis of the results wehave obtained, we get a stochastic dynamical system (RDS).And second, we estimate the solution of the stochastic damped wave equation byusing splitting technique. We fst separate the unbounded domain into two parts-inand out of a bounded ball, and then estimate the solution in each of them. In thisway we prove the existence of its absorbing set.At last, we prove the asymptotic compactness by using the results of the previoussection. Thus, we turn out the existence of random attractor for the stochastic dampedwave equation.The main work we did in this paper is that we generalize the results of theexistence of the random attractor for the stochastic damped wave equations frombounded domains to unbounded domains.