The existence of a global attractor for a Kuramoto-Sivashinsky type equation in 2D
Aslihan Demirkaya
Conference Publications 2009, 2009(Special): 198-207 doi: 10.3934/proc.2009.2009.198
We consider a variation of the Kuramoto-Sivashinsky Equation in space dimension two. We show that under some assumptions the equation is globally well-posed and posseses a global attractor in the periodic case. The analysis is based on the Lyapunov function approach, point dissipativeness and asymptotic compactness.
keywords: Attractors Kuramoto- Sivashinsky Equation
Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
Aslihan Demirkaya Milena Stanislavova
Discrete & Continuous Dynamical Systems - B 2019, 24(1): 197-209 doi: 10.3934/dcdsb.2018097

In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: $u_{tt}+u_{xxxx}+u-|u|^{p-1} u = 0$ when $p = 3$ and $p = 5$. For the standing wave solutions $u(x, t) = e^{iω t}\varphi_{ω}(x)$ we numerically illustrate their existence using variational approach. Our numerics illustrate the existence of both ground states and excited states. We also compute numerically the threshold value $ω^*$ which separates stable and unstable ground states. Next, we study the existence and linear stability of periodic traveling wave solutions $u(x, t) = φ_c(x+ct)$. We present numerical illustration of the theoretically predicted threshold value of the speed $c$ which separates the stable and unstable waves.

keywords: Nonlinear beam equation standing waves traveling waves spectral stability orbital stability

Year of publication

Related Authors

Related Keywords

[Back to Top]