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Limit behavior of nonlinear stochastic wave equations with singular perturbation
| 1. | Department of Mathematics, Nanjing University, Nanjing, 210093, China |
| 2. | School of Science, Nanjing University of Science & Technology, Nanjing, 210094, China |
$ \nu $utt$+u_t=$Δ$u+f(u)+$ε$\dot{W}$
on an open bounded domain $D\subset\R^n$, $1\leq n\leq 3$,, is studied in the sense of distribution for small $\nu, $ε$>0$. Here $\nu$ is the parameter that describes the singular perturbation. First, by a decomposition of Markov semigroup defined by (1), a stationary solution is constructed which describes the asymptotic behavior of solution from initial value in state space $H_0^1(D)\times L^2(D)$. Then a global measure attractor is constructed for (1). Furthermore under the case that the stochastic force is proportional to the square root of singular perturbation, that is ε$=\sqrt{\nu}$, we study the limit of the behavior of all the stationary solutions of (1) as $\nu\rightarrow 0$. It is shown that, by studying a continuity property on $\nu$ for the measure attractors of (1), any one stationary solution of the limit equation
$u_t=$Δ$u+f(u).$
is a limit point of a stationary solution of (1), as $\nu\rightarrow 0$.
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