1999 Large time behavior of solutions of higher order nonlinear dispersive equations of KdV type with weak nonlinearity
Nakao Hayashi, Tonatiuh Matos, Pavel I. Naumkin
Differential Integral Equations 12(1): 23-40 (1999). DOI: 10.57262/die/1367266992


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.


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Nakao Hayashi. Tonatiuh Matos. Pavel I. Naumkin. "Large time behavior of solutions of higher order nonlinear dispersive equations of KdV type with weak nonlinearity." Differential Integral Equations 12 (1) 23 - 40, 1999. https://doi.org/10.57262/die/1367266992


Published: 1999
First available in Project Euclid: 29 April 2013

zbMATH: 1022.35058
MathSciNet: MR1668529
Digital Object Identifier: 10.57262/die/1367266992

Primary: 35Q53
Secondary: 35B40

Rights: Copyright © 1999 Khayyam Publishing, Inc.


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Vol.12 • No. 1 • 1999
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