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January  2010, 13(1): 175-193. doi: 10.3934/dcdsb.2010.13.175

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

Received  January 2009 Revised  August 2009 Published  October 2009

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 $\nu\rightarrow 0$.

Citation: Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175
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