TY - JOUR T1 - Global Existence and Exponential Decay of Solutions of Generalized Kuramoto-Sivashinsky Equations 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 -

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'(u)| ≤ c|u|^{α-1} for some α > 1. We prove that if λ < \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) > 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.