(untitled) (Journal of Partial Differential Equations, 2005)
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 < 4π²l², 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 d^4dx^4 + λd²dx²), then the solution for small initial datum is global and exponentially decays to zero.