扰动KdV方程解的先验估计

Priori estimates for the solutions of perturbed Korteweg-de Vries Equation

对于带微扰的KdV方程ut+6uux+uxx=εR(u),(ε>0),在初值u0(x)∈C∞(-∞,+∞),当|x|→∞时指数衰减的条件下,分别构造出带两种不同扰动项的KdV方程的扰动孤立波解满足的能量关系式,并运用能量分析方法对扰动的孤立波解进行先验估计,得到如下结论:(1)R(u)=δ(tε)u,δ(s)∈C[0,+∞),δ(0)=0,时,解在-∞<x<+∞,0≤εt≤T内一致有界;(2)R(u)=-Δ(tε)uxxx,Δ(0)=0,Δ(s)∈C1[0,+∞),解在-∞<x<+∞,0≤tε≤T,0≤ε≤ε1内一致有界。

Energy equalities are constructed for the perturbed solitary wave solutions corresponding to two kinds of perturbations for the perturbed KdV equation ut＋6uux＋uxx=εR（u）,（ε〉0） under the condition that the initial data u0（x）∈C^∞（-∞＋∞） decay exponentially as ｜x｜→∞. Priori estimates of the bound of the solutions are obtained via the method of energy analysis： （1）if R（u）=δ（εt）u,δ（s）∈C[0,＋∞）,δ（0）=0 and δ（0） = 0, the solutions are uniformly bounded in the region -∞〈x〈＋∞,0≤εt≤T; （2） in the case of R（u）=-△（εt）uxxx,△（0）=0,△（s）∈C^1[0,＋∞）, the solutions are uniformly bounded in the region -∞〈x〈＋∞,0≤εt≤T,0≤ε≤ε1 for some positive small el.