Evolution of the Radius of Spatial Analyticity for the Dispersion Modified Degasperis-Procesi Equation

Here, the local well-posedness of the Cauchy problem for the dispersion modified b-equation with data in Sobolev spaces H^s(R) and analytic Gevrey spaces G^d,s(R) is proved for any s > 1/4. However, for b = 3, which is the modified Degasperis-Procesi equation, a sharper result is established. In this case, the equation behaves as a nonlocal perturbation of the Kordeweg-de Vries (KdV) equation and well-posedness is shown for s > -3/4. Furthermore, for b = 3, this equation possesses a twisted-L² conservation law. This yields an almost conservation law in the analytic Gevrey spaces G^d,0. Using this almost conservation law, global solutions are established and a lower bound, given by c/t ⁴/^{3 +} , for their radius of spatial analyticity is proved. Key ingredients in the proof of this result are the Paley-Wiener Theorem and bilinear estimates for the nonlinearity of the modified Degasperis-Procesi equation.