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Volume 18, Issue 2
Global Existence and Exponential Decay of Solutions of Generalized Kuramoto-Sivashinsky Equations

Shangbin Cui & Cuihua Guo

J. Part. Diff. Eq., 18 (2005), pp. 167-184.

Published online: 2005-05

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  • Abstract

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.

  • Keywords

Generalized Kuramoto-Sivashinsky equations initial-boundary value problem global existence exponential decay

  • AMS Subject Headings

35K35

  • Copyright

COPYRIGHT: © Global Science Press

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@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 = {

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.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5353.html} }
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 -

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.

Shangbin Cui and Cuihua Guo . (2005). Global Existence and Exponential Decay of Solutions of Generalized Kuramoto-Sivashinsky Equations. Journal of Partial Differential Equations. 18 (2). 167-184. doi:
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