一类广泛kdv方程Cauchy问题的整体解及渐近性

Global Solution and Asymptotic Behavior of for A Class of Generalized kdv Equations

本文讨论了一类广泛kdv方程(P)的Cauchy问题。对初值φ(x)及非线性项加以适当限制,在Sobolev空间框架下,用算子半群理论及不动点原理得到了整体解的存在唯一性和渐近性。

In this paper we discuss the following Cauchy problem for a class of generalized kdv equation: u_t+βD^3u+αD^2u+f(u)Du=0,(α<0),(x,t)∈R^1×R^1+t=0: u=φ(x), x∈R^1 Under the condition that the initial data φ(x) is small in the norm in Sobolev space H^(s+1)∩W^(s+y+2,1)(s≥N≥2 integer) and some restriction on non-linear term f(u) by using the semigroup theory and Banach fixed-point princple, we obtain the existence end uniqueness of Global solutions of the Cauchy problem in Sobolev space and asymptotic behavior of the solution as t→∞+.