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Limit behavior of nonlinear stochastic wave equations with singular perturbation

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  • Dynamical behavior of the following nonlinear stochastic damped wave equations

    $ \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 $\nu\rightarrow 0$.

    Mathematics Subject Classification: Primary: 60F10, 60H15; Secondary: 35Q55.

    Citation:

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