Forced oscillation of a damped BBM equation posed on whole line in low regularity spaces

In this manuscript, we would established in low regularity spaces $H^\ell, \ell\in [0,1)$, the existence and stability results of time-periodic solution of 1D Cauchy problem of forced damped Benjamin-Bona-Mahony equation (BBM). We use estimates from I-energy method to derive needed estimates in $H^\ell$ for the linearized problem, then convection term will be treated as perturbation of linear problem such that original Cauchy problem is solved.