The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity

It is proved herein that certain smooth, global solutions of a class of quasi-linear, dissipative wave equations have precisely the same leading order, long-time, asymptotic behavior as the solutions with the same initial data of the corresponding linearized equations. The solutions of the nonlinear equations are shown to be asymptotically self-similar with explicitly determined profiles. The equations considered have homogeneous nonlinearities and homogeneous dispersive and dissipative symbols. By relating these degrees of homogeneous to the leading order asymptotic behavior of the Fourier transform of the initial data near k= 0, different classes of long-time asymptotic behavior are characterized. These results cover the case where dissipation is not asymptotically negligible in comparison with dispersion, and where nonlinear effect are asymptotically negligible in comparison with linear effect, i.e., dissipation and dispersion. They always hold for solutions with "small" initial data. In most circumstances however a new a priori bound on certain negative homogeneous Sobolev norms of solutions is obtained, which implies that any solution, even one which is initially "large" will eventually satisfy the smallness condition, and hence will have the above described asymptotic behavior