GLOBAL EXISTENCE AND EXPONENTIAL DECAY OF SOLUTIONS OF GENERALIZED KURAMOTO-SIVASHINSKY EQUATIONS

摘要

We study the Dirichlet initial-boundary value problem of the general-ized Kuramoto-Sivashinsky equation ut + uxxx + λuxx + f(u)x = 0 on the interval[0, l]. The nonlinear function f satisfies the condition |f'(u)| ≤ c|u|α-1 for some α＞ 1. We prove that if λ＜ 4π2/l2, then the strong solution is global and exponentially decays to zero for any initial datum uo ∈ Ho2(0,l) if 1 ＜α≤ 7, and for small uo ∈ H2O(0,l) if α＞ 7. We then consider the equation ut + uxxxx + λuxx + μu + auxxx + bux = F(u,ux,uxx,uxxx). We prove that if F is twice differentiable, ▽2F is Lipschitz con- tinuous, and F(0) = ▽F(0) = 0, and if λ andμ satisfyμ + σ(λ) ＞ 0 (σ(λ)=the first eigenvalue of the operator d4/dx4+λ d2/dx2), then the solution for small initial datum is global and exponentially decays to zero.