@Article{JPDE-18-167,
author = {Shangbin Cui  and Cuihua Guo },
title = {Global Existence and Exponential Decay of Solutions of Generalized Kuramoto-Sivashinsky Equations},
journal = {Journal of Partial Differential Equations},
year = {2005},
volume = {18},
number = {2},
pages = {167--184},
abstract = {<p>We study the Dirichlet initial-boundary value problem of the general- ized Kuramoto-Sivashinsky equation u_t + u_{xxxx} + λu_{xx} + f(u)_ x = 0 on the interval [0, l]. The nonlinear function f satisfies the condition |f&#39;(u)| ≤ c|u|^{α-1} for some α &gt; 1. We prove that if λ &lt; \frac{4π²}{l²}, then the strong solution is global and exponentially decays to zero for any initial datum u_0 ∈ H²_0 (0, l) if 1 ‹ α ≤ 7, and for small u_0 ∈ H²_0 (0, l) if α › 7. We then consider the equation u_t + u_{xxxx} + λu_{xx} + μu + au_{xxx} + bu_x = F(u, u_x, u_{xx}, u_{xxx}). We prove that if F is twice differentiable, ∇²F is Lipschitz continuous, and F(0) = ∇F(0) = 0, and if λ and μ satisfy μ + σ(λ) &gt; 0 (σ(λ)=the first eigenvalue of the operator \frac{d^4}{dx^4} + λ\frac{d²}{dx²}), then the solution for small initial datum is global and exponentially decays to zero.</p>},
issn = {2079-732X},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jpde/5353.html}
}