Hm convergence rates of solutions of GBBM equations in multi-dimensional spaces

In this paper, we study the decay rates of the generalized Benjamin–Bona–Mahony equations in n-dimensional space. By using Fourier analysis for long wave and by applying the energy method for short wave, we obtain the Hm convergence rates of the solutions when the initial data are in the bounded subset of the phase space Hm(Rn)(n⩾3)Hm(Rn)(n⩾3). The optimal decay rates are obtained in our results and are found to be the same as the Heat equation.