TY - JOUR
T1 - Global Existence and Exponential Decay of Solutions of Generalized Kuramoto-Sivashinsky Equations
AU - Shangbin Cui  & Cuihua Guo 
JO - Journal of Partial Differential Equations
VL - 2
SP - 167
EP - 184
PY - 2005
DA - 2005/05
SN - 18
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5353.html
KW - Generalized Kuramoto-Sivashinsky equations
KW - initial-boundary value problem
KW - global existence
KW - exponential decay
AB - <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>