Differential and Integral Equations

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

Nakao Hayashi, Tonatiuh Matos, and Pavel I. Naumkin

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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.

Article information

Source
Differential Integral Equations, Volume 12, Number 1 (1999), 23-40.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367266992

Mathematical Reviews number (MathSciNet)
MR1668529

Zentralblatt MATH identifier
1022.35058

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions

Citation

Hayashi, Nakao; Matos, Tonatiuh; Naumkin, Pavel I. Large time behavior of solutions of higher order nonlinear dispersive equations of KdV type with weak nonlinearity. Differential Integral Equations 12 (1999), no. 1, 23--40. https://projecteuclid.org/euclid.die/1367266992


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