Asymptotic Solutions of Semilinear Stochastic Wave Equations
نویسنده
چکیده
Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in mean-square and the almost sure sense, are studied. Then, under some sufficient conditions, the existence of a unique invariant measure is proved. Two examples are presented to illustrate some applications of the theorems.
منابع مشابه
Continuous dependence on coefficients for stochastic evolution equations with multiplicative Levy Noise and monotone nonlinearity
Semilinear stochastic evolution equations with multiplicative L'evy noise are considered. The drift term is assumed to be monotone nonlinear and with linear growth. Unlike other similar works, we do not impose coercivity conditions on coefficients. We establish the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. As corollaries of ...
متن کاملAsymptotics of Solutions to Semilinear Stochastic Wave Equations
Large-time asymptotic properties of solutions to a class of semi-linear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponent...
متن کاملAsymptotic Behavior of Stochastic Wave Equations with Critical Exponents on R
The existence of a random attractor in H1(R3)×L2(R3) is proved for the damped semilinear stochastic wave equation defined on the entire space R 3. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the rand...
متن کاملErgodic Control of Semilinear Stochastic Equations and Hamilton-jacobi Equations
In this paper we consider optimal control of stochastic semilinear equations with linearly increasing drift and cylindrical noise. We show existence and uniqueness (up to an additive constant) of solutions to the stationary Hamilton-Jacobi equation associated with the cost functional given by the asymptotic average per unit time cost. As a consequence we nd the optimizing controls given in the ...
متن کاملAsymptotic behaviour of solutions of semilinear hyperbolic systems in arbitrary domains
In this paper the long time asymptotic behavior of solutions of semilinear symmetric hyperbolic system including Maxwell s equations and the scalar wave equation in an ar bitraty domain are investigated The possibly nonlinear damping term may vanish on a certain subset of the domain It is shown that the solution decays weakly to zero if and only if the initial state is orthogonal to all station...
متن کامل