Well-Posedness and Exponential Stability of the Kuramoto-Sivashinsky Equation in a Bounded Interval and With a Constant Boundary Time-Delay

• Boumediène Chentouf ^[1] ; Nejib Smaoui ^[1]
  1. [1] Kuwait University

     Kuwait University

     Kuwait

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this article, we consider a one-dimensional dispersive equation known in literature as Kuramoto-Sivashinsky (KS) equation. It models numerous physical phenomenon such as turbulent states in a distributed chemical reaction, flame propagation and waves in a viscous fluid. Motivated by practical reasons, a constant time-delay is assumed to occur in the boundary and then the main purpose is to study the effect of the presence of this delay on the behavior of the solutions to the KS equation.

    To do so, we first show that the problem is well-posed. Second, we prove that the solutions to the problem are exponentially stable in the energy space. Finally, our theoretical findings are ascertained by means of a thorough numerical analysis of the problem and several numerical simulations are provided. Our findings extend earlier results that neglected boundary-delay effects and, moreover, improve previous results by addressing a streamlined model under more natural boundary conditions.

• Referencias bibliográficas
  • 1. Adams, R.A.: Sobolev Spaces. Academic Press, New York-London (1975)
  • 2. Ammari, K., Chentouf, B., Smaoui, N.: A qualitative study and numerical simulations for a timedelayed dispersive equation. J. Appl. Math....
  • 3. Baudouin, L., Cerpa, E., Crépeau, E., Mercado, A.: Lipschitz stability in an inverse problem for the Kuramoto-Sivashinsky equation. Applicable...
  • 4. Biagioni, H.A., Bona, J.L., Iorio, R.J., Jr., Scialom, M.: On the Korteweg-de Vries-KuramotoSivashinsky equation. Adv. Differ. Equations...
  • 5. de A Capistrano-Filho, R., Chentouf, B., de Jesus, I.M.: Dynamic stability of the Kawahara equation under the effect of a boundary finite...
  • 6. Cazacu, C.M., Ignat, L.I., Pazoto, A.F.: Null-Controllability of the linear Kuramoto-Sivashinsky equation on star-shaped trees. SIAM J....
  • 7. Cerpa, E.: Null controllability and stablization of the linear Kuramoto-Sivashinsky equations, Commun. Pure. Appl. Anal. 9, 91–102 (2010)
  • 8. Cerpa, E.: Boundary control of Korteweg-de vries and Kuramoto-Sivashinsky PDEs. In: Baillieul, J., Samad, T. (eds.) Encyclopedia of systems...
  • 9. Cerpa, E., Mercado, A.: Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation. J. Differ. Equations...
  • 10. Cerpa, E., Mercado, A., Pazoto, A.: Null controllability of the stabilized Kuramoto-Sivashinsky system with one distributed control. SIAM...
  • 11. Cerpa, E., Guzmán, P., Mercado, A., Pazoto, A.: On the control of the linear Kuramoto-Sivashinsky equation. ESAIM: COCV 23, 165–194 (2017)
  • 12. Chang, H.C.: Nonlinear waves on liquid film surfaces-II. Flooding in a vertical tube. Chem. Eng. Sci. 41, 2463–2476 (1986)
  • 13. Chentouf, B.: Well-posedness and exponential stability results for a nonlinear Kuramoto-Sivashinsky equation with a boundary time-delay....
  • 14. Chentouf, B.: On the exponential stability of a nonlinear Kuramoto-Sivashinsky-Korteweg-de Vries equation with finite memory. Mediterr....
  • 15. Chentouf, B.: Qualitative analysis of the dynamic for the nonlinear Korteweg-de Vries equation with a boundary memory. Qual. Theory Dyn....
  • 16. Chentouf, B., Smaoui, N., Alalabi, A.: Nonlinear adaptive boundary control of the modified generalized Korteweg-de Vries-Burgers equation....
  • 17. Cohen, B.I., Krommes, J.A., Tang, W.M., Rosenbluth, M.N.: Nonlinear saturation of the dissipative trapped-ion mode by mode coupling. Nuclear...
  • 18. Coron, J.M., Lü, Q.: Fredholm transform and local rapid stabilization for a Kuramoto-Sivashinsky equation. J. Differ. Equations 259, 3683–3729...
  • 19. De Angelis, M.: A priori estimates for solutions of FitzHugh-Rinzel system. Meccanica 57, 1035–1045 (2022). https://doi.org/10.1007/s11012-022-01489-6
  • 20. Gao, P.: A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability...
  • 21. Gao, P.: Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evol. Equ. Control Theory 4, 281–296...
  • 22. Gao, P.: Global exact controllability of the trajectoris of the Kuramoto-Sivashinsky equation. Evol. Equ. Control Theory 9, 181–19 (2020)
  • 23. Gomes, S.N., Papageorgiou, D.T., Pavliotis, G.A.: Stabilizing nontrivial solutions of the generalized Kuramoto-Sivashinsky equation using...
  • 24. Guzmán, P., Marx, S., Cerpa, E.: Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control. IFAC Papers...
  • 25. Hu, C., Temam, R.: Robust control of the Kuramoto-Sivashinsky equation. Dyn. Contin. Discrete Impuls. Syst. Ser. B, Appl. Algorithms 8,...
  • 26. Kang, W., Fridman, E.: Distributed sampled-data control of Kuramoto-Sivashinsky equation. Automatica 95, 514–524 (2018)
  • 27. Kobayashi, T.: Adaptive stabilization of the Kuramoto-Sivashinsky equation, International. J Systems Sci. 33, 175–180 (2002)
  • 28. Kuramoto, Y., Tsuzuki, T.: On the formation of dissipative structures in reaction-diffusion systems. Progr. Theoret. Phys. 54, 687–699...
  • 29. Kuramoto, Y.: Diffusion-induced chaos in reaction systems. Prog. Theor. Phys. Supplements 64, 346–367 (1978)
  • 30. Liu, W.J., Krstic, M.: Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation, Nonlinear Anal. Theory, Methods...
  • 31. Lou, Y., Christofides, P.: Optimal actuator/sensor placement for nonlinear control of the KuramotoSivashinsky equation. IEEE Trans. Control...
  • 32. Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks....
  • 33. Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of the KuramotoSivashinsky equations: Nonlinear stability and...
  • 34. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. SpringerVerlag, New York (1983)
  • 35. Russell, J.S.: Experimental researches into the laws of certain hydrodynamical phenomena that accompany the motion of floating bodies...
  • 36. Sakthivel, R., Ito, H.: Nonlinear robust boundary control of the Kuramoto-Sivashinsky equation. IMA J. of Math. Control and Information...
  • 37. Shangbin, C., Cuihua, G.: Global existence and exponential decay of solutions of generalized Kuramoto-Sivashinsky equations. J. Partial...
  • 38. da Silva, P.N., Vasconcellos, C.F.: On the Kuramoto-Sivashinsky system in a bounded domain. Far East J. of Mathematical Sci. 71, 47–65...
  • 39. Sivashinsky, G.: Nonlinear analysis for hydrodynamic instability in Laminar flames. Derivation of basic equations, Acta Astronautica 4,...
  • 40. Sivashinsky, G.I.: On flame propagation under conditions of stoichiometry. SIAM J. Applied Math. 39, 67–82 (1980)
  • 41. Smaoui, N., Chentouf, B., Alalabi, A.: Modelling and nonlinear boundary stabilization of the modified generalized Korteweg-de Vries-Burgers...
  • 42. Smaoui, N., Chentouf, B., Alalabi, A.: Boundary linear stabilization of the modified generalized Korteweg-de Vries-Burgers equation. Adv....
  • 43. Smaoui, N., Zribi, M.: Controlling the dynamics of the Kuramoto-Sivashinsky equation. Int. J. Pure Appl. Math. 7, 271–294 (2003)
  • 44. Sun, B.,Wu,M.X.: Optimal boundary control of a coupled system consisting of Kuramoto-SivashinskyKorteweg-de Vries and heat equations....
  • 45. Topper, J., Kawahara, T.: Approximate equations for long nonlinear waves on a viscous fluid. J. Phys. Soc. Japan 44, 663–666 (1978)
  • 46. Vasconcellos, C.F., da Silva, P.N.: Exact Controllability and Stabilization for Kuramoto-Sivashinsky System, Proceeding Series of the...
  • 47. Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var....
  • 48. Zhu, N., Liu, Z., Zhao, K.: Non blowup of a generalized Boussinesq-Burgers system with nonlinear dispersion relation and large data. Physica...
  • 49. Zhu, N., Qu, W.: Existence and bifurcation of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion....