  Based on the nonlinear steepest descent method of Deift and Zhou for
oscillatory Riemann--Hilbert problems and the Dbar approach, the long-time
asymptotic behavior of solutions to the fifth-order modified Korteweg-de Vries
equation on the line is studied in the case of initial conditions that belong
to some weighted Sobolev spaces. Using techniques in Fourier analysis and the
idea of $I$-method, we give its global well-posedness in lower regularity
Sobolev spaces, and then obtain the asymptotic behavior in these spaces with
weights.

Analysis of PDEs