科研之友
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摘要
Dynamical behavior of the following nonlinear stochastic damped wave equations.
nu u(tt) + u(t) = Delta u + f(u) + epsilon(W) over dot (1)
on an open bounded domain D subset of R(n), 1 <= n <= 3, is studied in the sense of distribution for small nu, epsilon > 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) x 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 epsilon = root nu, we study the limit of the behavior of all the stationary solutions of (1) as nu -> 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) = Delta u + f(u). (2)
is a limit point of a stationary solution of (1),a nu -> 0.
