@Article{JPDE-32-304,
author = {Caraballo , TomásPiña , Nicolás and Muñoz , Jaime},
title = {Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise},
journal = {Journal of Partial Differential Equations},
year = {2020},
volume = {32},
number = {4},
pages = {304--325},
abstract = {<p>The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic
dynamics when zero is not a solution of the equation by using an appropriate cocycle
and random dynamical system. The stability of mild solutions is proved in both cases
of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the
existence of a non-trivial stationary solution which is exponentially stable, by using a
general random fixed point theorem for general cocycles. In this case, the stationary
solution is generated by the composition of random variable and Wiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and
prove the existence of a random fixed point exponentially attracting every path.</p>},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v32.n4.2},
url = {http://global-sci.org/intro/article_detail/jpde/13611.html}
}