扰动KdV方程解的先验估计

对于带微扰的KdV方程u_t+6uu_x+u_xx=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(εt)u, δ(s)∈C［0,+∞)，δ(0)=0，时，解在-∞＜x＜+∞，0≤εt≤T内一致有界；(2)R(u)=-Δ(εt)u_xxx，Δ(0)=0，Δ(s)∈C¹［0,+∞), 解在-∞＜x＜+∞，0≤εt≤T，0≤ε≤ε₁内一致有界。

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

Energy equalities are constructed for the perturbed solitary wave solutions corresponding to two kinds of perturbations for the perturbed KdV equation u_t+6uu_x+u_xx=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,+∞) and δ(0)=0, the solutions are uniformly bounded in the region -∞＜x＜+∞，0≤εt≤T; (2) in the case of R(u)=-Δ(εt)u_xxx, Δ(0)=0, Δ(s)∈C¹［0,+∞), the solutions are uniformly bounded in the region -∞＜x＜+∞, 0≤εt≤T, 0≤ε≤ε₁ for some positive small ε₁.