Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise (Journal of Partial Differential Equations, 2020)
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 asymptoticdynamics when zero is not a solution of the equation by using an appropriate cocycleand random dynamical system. The stability of mild solutions is proved in both casesof Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate theexistence of a non-trivial stationary solution which is exponentially stable, by using ageneral random fixed point theorem for general cocycles. In this case, the stationarysolution 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 andprove the existence of a random fixed point exponentially attracting every path.