On the asymptotic behavior of solutions of generalized Korteweg-de Vries equations (Q1262453)

The author handles the generalized Korteweg-de Vries equation \[ (GKdV)\quad u_ t+u_{xxx}+p| u|^{p-1}u_ x=0. \] In particular he is interested in the asymptotic behaviour of the solution of the corresponding Cauchy problem with initial-data \(\psi \in L^ 2({\mathbb{R}})\). These are the main results: Theorem 1. Let u(x,t) be a solution of (GKdV) with \(u(x,0)=\psi \in L^ 2({\mathbb{R}}),\) \(\| \psi \| \neq 0\). Then there does not exist any solution v(x,t) of the Airy equation \((A)\quad v_ t+v_{xxx}=0,\) with \(v(x,0)=\phi (x)\in L^ 1({\mathbb{R}}),\quad \int \phi (x)dx=0,\) such that \(\| u(t)-v(t)\|_{L^ 2({\mathbb{R}})}\to 0\) as \(t\to \infty.\) Theorem 2. Let u(x,t) be as in Theorem 1 and \(\psi\) satisfies in addition \(\int \psi (x)dx=0\). Then there does not exist any solution v(x,t) of the Airy equation with \(v(x,0)=\phi (x)\in L^ 1({\mathbb{R}})\) and \[ \| u(t)-v(t)\|_{L^ 2({\mathbb{R}})}+\| u(t)-v(t)\|_{L^ 1({\mathbb{R}})}\to 0\quad as\quad t\to +\infty. \]