Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation {u(tt) + 2 alpha u(t) - beta u(xx) = -lambda vertical bar u vertical bar(sigma)u, x is an element of Omega, t > 0, u(0, x) = phi (x), u(t)(0, x) = psi (x), x is an element of Omega, where Omega = [-pi, pi], alpha, beta, lambda, sigma > 0. We prove that if the initial data phi is an element of H(1) and psi is an element of L(2), then there exists a unique solution u (t, x) is an element of C ([0, infinity); H(1)) of the periodic problem which has the time decay estimates parallel to u(t)parallel to(L infinity) <= C < t >(-1/sigma), parallel to partial derivative(t)u (t)parallel to(L infinity) <= C < t >(-1/sigma-1/2) for all t > 0. Moreover under some additional conditions we find the asymptotic formulas for the solutions.