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

We consider the periodic problem for the nonlinear damped wave equation with pumping and convective nonlinearity {v(tt) + 2 alpha v(t) - nu(xx) = v = partial derivative(x) (v(x)(3)), x is an element of Omega, t > 0 v(0, x) > (phi) over tilde (x) = v(t) (0, x) = (psi) over tilde (x), x is an element of Omega, where alpha > 0, Omega = [-pi, pi]. We study the solutions, which satisfy the periodic boundary conditions v (t, x) = v (t, 2 pi + x) for all x is an element of R and t > 0, with the 2 pi - periodic initial data (phi) over tilde (x) and (psi) over tilde (x). Our aim in the present paper is to find the large time asymptotics for solutions to the periodic problem for the nonlinear damped wave equation (1.1) carefully studying the behavior of the first harmonics of the solution and applying the energy type estimates. We prove the following asymptotics for the solutions v (t, x) = (phi) over tilde (0) + 1/2 alpha(psi) over tilde (0) + epsilon/root 1 + 2b epsilon(2)t + O((1 + epsilon(2) t)(-1))| as t -> 8 uniformly with respect to x is an element of, where (phi) over tilde (0) = integral(Omega) (phi) over tilde (x) dx, (psi) over tilde (0) = integral(Omega) (psi) over tilde (x) dx, epsilon = vertical bar integral(Omega) e(tx)(phi) over tilde (x)dx vertical bar, b = 3/2 alpha.