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Asymptotic property of solutions to the random cauchy problem for wave equations

✍ Scribed by Isamu Dôku

Publisher
Springer
Year
1986
Tongue
English
Weight
378 KB
Volume
11
Category
Article
ISSN
0377-9017

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✦ Synopsis

Based on the existence and umqueness theorem for random wave equations, we consider asymptotic behaviors of solutions to the random initial value problem. We describe conditions for the equlpartition of stochastic energy (or SE for short) by making use of the random spectral theory and, according to Goldstem's semigroup method, we prove the asymptotically equipartitioned SE theorem and the so-called virial theorem of classical mechanics, and also study the probabilistic characterization of the conditions for equipartition. In addition, we show at last the equipartition of SE from a finite time onwards.

The uniqueness takes place in the sense that sup, II w(t)II = 0 holds with probability 1 if w(t) is a difference process of two solution processes [6]. We state the random

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## Abstract Let __D__ ⊂ ℝ^__n__^ be a bounded domain with piecewise‐smooth boundary, and __q__(__x__,__t__) a smooth function on __D__ × [0, __T__]. Consider the time‐like Cauchy problem magnified image magnified image Given __g__, __h__ for which the equation has a solution, we show how to approxi