Large time behavior of solutions of higher order nonlinear dispersive equations of KdV type with weak nonlinearity

Abstract

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the higher-order dispersive equations of Korteweg-de Vries type with weak nonlinearity (wKdV): $$ u_t + \partial_x f(u) + \mathcal{K} u = 0, $$ where $x, t \in \mathbf {R}$, $f(u) = |u|^{\rho -1}u$ if $\rho > \nu$ or $f(u) = u^{\rho }$ if $\rho > \nu$ is integer, the operator $\mathcal{K} $ is a pseudodifferential operator with a homogeneous and conservative symbol $K(p)$ of order $\nu > 3$, namely, $\mathcal{K}u = \mathcal{F}^{-1} K(p) \hat u(p), K(p) = - \frac{i}{\nu} |p|^{\nu - 1} p, $ $\mathcal{F}\phi$ or $\hat \phi$ is the Fourier transformation of $\phi$ and $\mathcal{F}^{-1} \phi $ is the inverse Fourier transformation of $\phi$. If the power $\rho$ of the nonlinearity is greater than $\nu$, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solution $u(t)$ satisfies the decay estimate $$ \|u(t)\|_{L^\beta} \le C(1 + t)^{-\frac{1}{\nu}(1-\frac{1}\beta)} \quad \text{for} \quad \beta \in (\frac{2\nu -2}{\nu - 2},\infty], $$ $$ \|uu_x(t)\|_{L^\infty} \le Ct^{-2/\nu}(1+t)^{-1/\nu} $$ and using these estimates we prove the existence of the scattering state $u_+\in L^2$ such that $$ \|u(t) - U(t)u_+\|_{L^2} \le Ct^{-\frac{\rho -\nu}{\nu}} \quad \text{and} \quad \|u(t) - U(t)u_+\|_{L^\infty} \le Ct^{-\frac{1+\rho -\nu}{\nu}} $$ for any small initial data belonging to the weighted Sobolev space $H^{1,1} = \{ \phi \in L^2; \|(1+|x|^2)^{1/2}(1-\partial_x^2)^{1/2} \phi\|_{L^2}<\infty\}$, where $U(t)$ is the free evolution group, associated with corresponding linear equation.