Well-posedness Of Fractional Benjamin-Bona-Mahony Equation

In this paper,we are interested in the behaviour of the solutions of the deter-ministic fractional Benjamin-Bona-Mahony equation:(?)tu +(?)xu + u(?)xu? D?(?)tu ? 0.where ?>0 and D? is defined by Fourier transformation(?)(?)?|?|?(?)(?),for all ?>0.When ?>1,we prove that the Cauchy problem is global well-posed in H?/2(R).Then,we prove that the Cauchy problem is local well-posed in H?(R),r>max,(1,3/2-?},where 0<?<1.Moreover,we give the decay rate of the nonlinear equation.In the last chapter,we illustrate numerically the weak dispersion of solution for the case ?? 1 and the explosion of solution for the case 0<?<1/3.