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 $H^{\mathrm{\ell }},\mathrm{\ell }\in [0,1)$, 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 $H^{\mathrm{\ell }}$ for the linearized problem, then convection term will be treated as perturbation of linear problem such that original Cauchy problem is solved.